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Research ArticleGeneral Decay and Blow-Up of Solutions for a System ofViscoelastic Equations of Kirchhoff Type with Strong Damping

Wenjun Liu Gang Li and Linghui Hong

College of Mathematics and Statistics Nanjing University of Information Science and Technology Nanjing 210044 China

Correspondence should be addressed to Wenjun Liu wjliunuisteducn

Received 8 May 2013 Revised 28 September 2013 Accepted 16 November 2013 Published 4 February 2014

Academic Editor William Ziemer

Copyright copy 2014 Wenjun Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping isconsidered We first establish two blow-up results one is for certain solutions with nonpositive initial energy as well as positiveinitial energy by exploiting the convexity technique the other is for certain solutions with arbitrarily positive initial energy basedon the method of Li and Tsai Then we give a decay result of global solutions by the perturbed energy method under a weakerassumption on the relaxation functions

1 Introduction

In this work we investigate the following system of viscoelas-tic equations of Kirchhoff type

119906119905119905minus119872(nabla119906

2

2) Δ119906 + int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) d119904 minus Δ119906

119905

= 1198911(119906 V) (119909 119905) isin Ω times (0infin)

V119905119905minus119872(nablaV2

2) ΔV + int

119905

0

1198922(119905 minus 119904) ΔV (119904) d119904 minus ΔV

119905

= 1198912(119906 V) (119909 119905) isin Ω times (0infin)

119906 (119909 119905) = 0 V (119909 119905) = 0

(119909 119905) isin 120597Ω times [0infin)

119906 (119909 0) = 1199060(119909) 119906

119905(119909 0) = 119906

1(119909) 119909 isin Ω

V (119909 0) = V0(119909) V

119905(119909 0) = V

1(119909) 119909 isin Ω

(1)

where Ω sub R119899 119899 ge 1 is a bounded domain with smoothboundary 120597Ω119872 is a positive locally Lipschitz functionand 119892

119894(sdot) R+ rarr R+ (119894 = 1 2) (119891

1 1198912) R2 rarr R2 are

given functions to be specified later

To motivate our work let us recall some previous resultsregarding viscoelastic equations of Kirchhoff type The fol-lowing problem

119906119905119905minus119872(nabla119906

2

2) Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 + ℎ (119906119905) = 119891 (119906)

(119909 119905) isin Ω times (0infin)

119906 = 0 (119909 119905) isin 120597Ω times [0infin)

119906 (119909 0) = 1199060(119909) 119906

119905(119909 0) = 119906

1(119909) 119909 isin Ω

(2)

is a model to describe the motion of deformable solidsas hereditary effect is incorporated It was first studiedby Torrejon and Yong [1] who proved the existence of aweakly asymptotic stable solution for large analytical datumLater Munoz Rivera [2] showed the existence of globalsolutions for small datum and the total energy decays to zeroexponentially under some restrictions Then Wu and Tsai[3] treated problem (2) for ℎ(119906

119905) = minusΔ119906

119905and proved the

global existence decay and blow-up with suitable conditionson initial data They obtained the blow-up properties oflocal solution with small positive initial energy by the directmethod of [4] To obtain the decay result they assumed thatthe nonnegative kernel 1198921015840(119905) le minus119903119892(119905) forall119905 ge 0 for some 119903 gt

0 This energy decay result was recently improved by Wu

Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 284809 21 pageshttpdxdoiorg1011552014284809

2 Journal of Function Spaces

in [5] under a weaker condition on 119892 (ie 1198921015840(119905) le 0 for119905 ge 0) For a single wave equation of Kirchhoff type withoutthe viscoelastic term we refer the reader to Matsuyama andIkehata [6] and Ono [7ndash10]

Many results concerning local existence global existencedecay and blow-up of solutions for a system of waveequations of Kirchhoff type without viscoelastic terms (ie119892119894

= 0 119894 = 1 2) have also been extensively studiedFor example Park and Bae [11] considered the system ofwave equations with nonlinear dampings for 119891

1(119906 V) =

120583|119906|119902minus1

119906 and 1198912(119906 V) = 120583|V|119902minus1V 119902 ge 1 and showed the

global existence and asymptotic behavior of solutions undersome restrictions on the initial energy Later Benaissa andMessaoudi [12] discussed blow-up properties for negativeinitial energy Recently Wu and Tsai [13] studied the system(1) for 119892

119894= 0 (119894 = 1 2) Under some suitable assumptions

on 119891119894(119894 = 1 2) they proved local existence of solutions by

applying the Banach fixed point theorem and the blow-up ofsolutions by using the method of Li and Tsai in [4] wherethree different cases on the sign of the initial energy 119864(0) areconsidered

In the case of 119872 equiv 1 and in the presence of viscoelasticterm (ie 119892 = 0) Cavalcanti et al [14] studied the equationthat was subject to a locally distributed dissipation

119906119905119905minus Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 + 119886 (119909) 119906119905+ |119906|120574119906 = 0


(3)

with the same initial and boundary conditions as that of (2)and proved an exponential decay rate This work extendedthe result of Zuazua [15] in which he considered (3) with119892 = 0 and the localized linear damping By using thepiecewise multipliers method Cavalcanti and Oquendo [16]investigated the equation

119906119905119905minus 1198960Δ119906 + int

119905

0

div [119886 (119909) 119892 (119905 minus 119904) nabla119906 (119904)] d119904

+ 119887 (119909) ℎ (119906119905) + 119891 (119906) = 0 (119909 119905) isin Ω times (0infin)

(4)

with the same initial and boundary conditions as that of (2)Under the similar conditions on the relaxation function 119892 asabove and 119886(119909) + 119887(119909) ge 120575 gt 0 for all 119909 isin Ω they improvedthe results of [14] by establishing exponential stability forexponential decay function 119892 and linear function ℎ andpolynomial stability for polynomial decay function 119892 andnonlinear function ℎ respectively

Concerning blow-up results Messaoudi [17] consideredthe equation

119906119905119905minus Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 + 119886119906119905

1003816100381610038161003816119906119905

1003816100381610038161003816

119898minus2

= 119887|119906|119903minus2

119906


(5)

He proved that any weak solution with negative initial energyblows up in finite time if 119903 gt 119898 and

int

infin

0

119892 (119904) d119904 le 119903 minus 2

119903 minus 2 + 1119903

(6)

while exists globally for any initial data in the appropriatespace if 119898 ge 119903 This result was improved by the same authorin [18] for positive initial energy under suitable conditions on119892 119898 and 119903 Recently Liu [19] studied the equation

119906119905119905minus Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 minus 120596Δ119906119905+ 120583119906119905= |119906|119903minus2

119906


(7)

with the same initial and boundary condition as that of (2)By virtue of convexity technique and supposing that

int

infin

0

119892 (119904) d119904 le 119903 minus 2

119903 minus 2 + 1 [(1 minus120575)

2

119903 + 2120575 (1 minus120575)]

(8)

where 120575 = max0 120575 he proved that the solution with

nonpositive initial energy as well as positive initial energyblows up in finite time

We should mention that the following system

119906119905119905minus Δ119906 + int

119905

0

1198921(119905 minus 120591) Δ119906 (120591) d120591 + 1003816

100381610038161003816119906119905

1003816100381610038161003816

119898minus1

119906119905= 1198911(119906 V)

(119909 119905) isin Ω times (0 119879)

V119905119905minus ΔV + int

119905

0

1198922(119905 minus 120591) ΔV (120591) d120591 + 1003816

100381610038161003816V119905

1003816100381610038161003816

120574minus1V119905= 1198912(119906 V)

(119909 119905) isin Ω times (0 119879)

119906 (119909 119905) = 0 V (119909 119905) = 0

(119909 119905) isin 120597Ω times [0 119879)

119906 (119909 0) = 1199060(119909) 119906

119905(119909 0) = 119906

1(119909) 119909 isin Ω

V (119909 0) = V0(119909) V

119905(119909 0) = V

1(119909) 119909 isin Ω

(9)

was considered by Han and Wang in [20] where Ω is abounded domain with smooth boundary 120597Ω in R119899 119899 =

1 2 3Under suitable assumptions on the functions119892119894 119891119894(119894 =

1 2) the initial data and the parameters in the above problemestablished local existence global existence and blow-upproperty (the initial energy 119864(0) lt 0) This latter blow-upresult has been improved byMessaoudi and Said-Houari [21]into certain solutions with positive initial energy RecentlyLiang and Gao in [22] investigated the following problem

119906119905119905minus Δ119906 + int

119905

0

1198921(119905 minus 120591)Δ119906 (120591) d120591 minus Δ119906

119905= 1198911(119906 V)

(119909 119905) isin Ω times (0 119879)


119905

0

1198922(119905 minus 120591)ΔV (120591) d120591 minus ΔV

119905= 1198912(119906 V)

(119909 119905) isin Ω times (0 119879)

(10)

with the same initial and boundary conditions as that of (9)Under suitable assumptions on the functions 119892

119894 119891119894(119894 = 1 2)

Journal of Function Spaces 3

and certain initial data in the stable set they proved that thedecay rate of the solution energy is exponential Converselyfor certain initial data in the unstable set they proved thatthere are solutions with positive initial energy that blow upin finite time It is also worth mentioning the work [23] inwhich we studied system (1) Under suitable assumptions onthe functions 119892

119894 119891119894(119894 = 1 2) and certain initial conditions

we showed that the solutions are global in time and the energydecays exponentially For other papers related to existenceuniform decay and blow-up of solutions of nonlinear waveequations we refer the reader to [14 24ndash29] for existence anduniform decay to [17 30ndash34] for blow-up and to [35ndash40] forthe coupled system To the best of our knowledge the generaldecay and blow-up of solutions for systems of viscoelasticequations of Kirchhoff type with strong damping have notbeen well studied

Motivated by the above mentioned research we con-sider in the present work the coupled system (1) withnonzero 119892

119894(119894 = 1 2) and nonconstant 119872(119904) We note that

in such a coupled system case we should overcome the addi-tional difficulties brought by the treatment of the nonlinearcoupled terms We first establish two blow-up results one isfor certain solutions with nonpositive initial energy as wellas positive initial energy the other is for certain solutionswith arbitrarily positive initial energy Then we give a decayresult of global solutions under a weaker assumption on therelaxation functions 119892

119894(119905) (119894 = 1 2)

This paper is organized as follows In the next section wepresent some assumptions notations and known results andstate the main results Theorems 4 5 6 and 7 The two blow-up results Theorems 5 and 6 are proved in Sections 3 and4 respectively Section 5 is devoted to the proof of the decayresultmdashTheorem 7

2 Preliminaries and Main Result

In this section we present some assumptions notations andknown results and state the main results First we make thefollowing assumptions(A1) 119872(119904) is a positive locally Lipschitz function for 119904 ge

0 with the Lipschitz constant 119871 satisfying

119872(119904) ge 1198980gt 0 (11)

(A2) 119892119894(119905) [0infin) rarr (0infin) (119894 = 1 2) are strictly

decreasing 1198621 functions such that

1198980minus int

infin

0

119892119894(119904) d119904 = 119897

119894gt 0 (12)

(A3) There exist two positive differentiable functions 1205851(119905)

and 1205852(119905) such that

1198921015840

119894(119905) le minus120585

119894(119905) 119892119894(119905) (119894 = 1 2) for 119905 ge 0 (13)

and 120585119894(119905) satisfies

1205851015840

119894(119905) le 0 forall119905 gt 0 int

+infin

0

120585119894(119905) d119905 = infin (119894 = 1 2)

(14)

(A4) We make the following extra assumption on 119872

2 (119901 + 2)119872 (119904) minus 2119872 (119904) 119904 ge 2 (119901 + 1)1198980119904 forall119904 ge 0 (15)

where 119872(119904) = int

119904

0119872(120591)d120591

Remark 1 It is clear that 119872(119904) = 1198980+ 1198981119904120574 (which occurs

physically in the study of vibrations of damped flexible spacestructures in a bounded domain in R119899) satisfies (A4) for 119904 ge0 1198980gt 0 119898

1ge 0 120574 ge 0 as long as 119901 + 1 gt 120574 Indeed by

straightforward calculations we obtain

2 (119901 + 2)119872 (119904) minus 2119872 (119904) 119904

= 2 (119901 + 2) (1198980119904 +

1198981

120574 + 1

119904120574+1

) minus 21198980119904 minus 2119898

1119904120574+1

= 2 (119901 + 1)1198980119904 +

2 (119901 + 1 minus 120574)

120574 + 1

1198981119904120574+1

ge 2 (119901 + 1)1198980119904

(16)

Next we introduce some notations Consider

(119892119894 nabla119908) (119905) = int

119905

0

119892119894(119905 minus 119904) nabla119908 (119905) minus nabla119908 (119904)

2

2d119904

119897 = min 1198971 1198972

1198920(119905) = max 119892

1(119905) 1198922(119905) forall119905 ge 0

sdot119902= sdot119871119902(Ω)

1 le 119902 le infin

(17)

the Hilbert space 1198712(Ω) endowed with the inner product

(119906 V) = int

Ω

119906 (119909) V (119909) d119909 (18)

and the functions 1198911(119906 V) and 119891

2(119906 V) (see also [21])

1198911(119906 V) = [119886|119906 + V|2(119901+1) (119906 + V) + 119887|119906|

119901119906|V|119901+2]

1198912(119906 V) = [119886|119906 + V|2(119901+1) (119906 + V) + 119887|119906|

119901+2|V|119901V]

(19)

where 119886 119887 gt 0 are constants and 119901 satisfies

minus 1 lt 119901 if 119899 = 1 2

minus 1 lt 119901 le

(3 minus 119899)

(119899 minus 2)

if 119899 ge 3

(20)

One can easily verify that

1199061198911(119906 V) + V119891

2(119906 V) = 2 (119901 + 2) 119865 (119906 V) forall (119906 V) isin R

2

(21)

where

119865 (119906 V) =1

2 (119901 + 2)

[119886|119906 + V|2(119901+2) + 2119887|119906V|119901+2] (22)

Then we give two lemmas which will be used throughoutthis work


Lemma 2 (Sobolev-Poincare inequality [41]) If 2 le 120588 le

2119899(119899 minus 2) then

119906120588le 119862nabla119906

2for 119906 isin 119867

1

0(Ω) (23)

holds with some constant 119862

Lemma 3 ([21 Lemma 32]) Assume that (20) holds Thenthere exists 120578

1gt 0 such that for any (119906 V) isin (119867

1

0(Ω) cap

1198672(Ω)) times (119867

1

0(Ω) cap 119867

2(Ω)) one has

119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

le 1205781(1198971nabla1199062

2+ 1198972nablaV22)

119901+2

(24)

We now state a local existence theorem for system (1)whose proof follows the arguments in [3 13]

Theorem 4 Suppose that (20) (1198601) and (1198602) hold andthat 119906

0 V0

isin 1198671

0(Ω) cap 119867

2(Ω) and 119906

1 V1

isin 1198712(Ω) Then

problem (1) has a unique local solution

119906 V isin 119862 ([0 119879) 1198671

0(Ω) cap 119867

2(Ω))

119906119905 V119905isin 119862 ([0 119879) 119871

2(Ω)) cap 119871

2([0 119879) 119867

1

0(Ω))

(25)

for some 119879 gt 0 Moreover at least one of the followingstatements is valid

(1) 119879 = infin

(2) lim119905rarr119879

minus

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ Δ119906

2

2+ ΔV2

2) = infin

(26)

The energy associated with system (1) is given by

119864 (119905) =

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1

2

(int

119905

0

1198921(119904) d119904) nabla119906

2

2

minus

1

2

(int

119905

0

1198922(119904) d119904) nablaV2

2minus int

Ω

119865 (119906 V) d119909

+

1

2

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2

119872(nabla1199062

2) +

1

2

119872(nablaV22) for 119905 ge 0

(27)

As in [5] we can get

1198641015840(119905)

= minus (1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2) minus

1

2

[1198921(119905) nabla119906

2

2+ 1198922(119905) nablaV2

2]

+

1

2

[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)] le 0 forall119905 ge 0

(28)

Then we have

119864 (119905) = 119864 (0) minus int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

minus

1

2

int

119905

0

[1198921(120591) nabla119906 (120591)

2

2+ 1198922(120591) nablaV (120591)2

2] d120591

+

1

2

int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

(29)

We introduce

1198611= 12057812(119901+2)

1 120572

lowast= 119861minus(119901+2)(119901+1)

1

1198641= (

1

2

minus

1

2 (119901 + 2)

)1205722

lowast

(30)

where 1205781is the optimal constant in (24)

Our first result is concerned with the blow-up for certainlocal solutions with nonpositive initial energy as well aspositive initial energy

Theorem 5 Assume that (1198601)-(1198602) (1198604) and (20) hold Let(119906 V) be the unique local solution to system (1) For any fixed120575 lt 1 assuming that we can choose (119906

0 V0) (1199061 V1) satisfy

119864 (0) lt 1205751198641 (119897

1

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

gt 120572lowast (31)

Suppose further that

int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus120575)

2

(119901 + 2) +120575 (1 minus

120575)]

(119894 = 1 2)

(32)

where 120575 = max0 120575 then 119879 lt infin

Our second result shows that certain local solutions witharbitrarily positive initial energy can also blow up

Theorem 6 Suppose that (A1)-(A2) (A4) (20) and

int

infin

0

119892119894(119904) d119904 le

2 (119901 + 1)1198980

2 (119901 + 1) + 12 (119901 + 2)

(119894 = 1 2) (33)


hold Assume further that 1199060 V0isin 1198671

0(Ω)cap119867

2(Ω) and 119906

1 V1isin

1198712(Ω) and satisfy

0 lt 119864 (0)

lt min

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

radic119901 + 3

(119901 + 2) (radic119901 + 3 minus radic119901 + 1)

int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(34)

then the solution of problem (1) blows up at a finite time 119879lowast inthe sense of (119) belowMoreover the upper bounds for119879lowast canbe estimated by

119879lowastle 2(3119901+5)2(119901+1)

(119901 + 1) 119888

2radic1198861

1 minus [1 + 119888ℎ (0)]minus1(119901+1)

(35)

where ℎ(0) = [11990602

2+ V02

2+ 1198791(nabla11990602

2+ nablaV

02

2)]minus(119901+1)2

and 1198861is given in (116) below

Finally we state the general decay result For conveniencewe choose especially 119872(119904) = 119898

0+ 1198981119904120574 1198980gt 0 119898

1ge 0

and 120574 ge 0 Then the energy functional 119864(119905) defined by (27)becomes

119864 (119905) = 119864 (119906 (119905) V (119905))

=

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(36)

Theorem 7 Suppose that (20) (1198601)-(1198602) and (1198603) holdand that (119906

0 1199061) isin (119867

1

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and (V

0 V1) isin

(1198671

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and satisfy 119864(0) lt 119864

1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (37)

Then for each 1199050gt 0 there exist two positive constants 119870 and

119896 such that the energy of (1) satisfies

119864 (119905) le 119870119890minus119896int119905

1199050120585(119904)d119904

119905 ge 1199050 (38)

where 120585(119905) = min1205851(119905) 1205852(119905)

To achieve general decay result we will use a Lyapunovtype technique for some perturbation energy following themethod introduced in [42] This result improves the one inLi et al [23] in which only the exponential decay rates areconsidered

3 Blow-Up of Solutions with InitialData in the Unstable Set

In this section we prove a finite time blow-up result for initialdata in the unstable set We need the following lemmas

Lemma 8 Suppose that (20) (1198601) (1198602) and (1198604) hold Let(119906 V) be the solution of system (1) Assume further that 119864(0) lt1198641and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12


Then there exists a constant 1205723gt 120572lowastsuch that

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 for 119905 isin [0 119879) (40)

Proof We first note that by (27) (24) and the definition of1198611 we have

119864 (119905) ge

1

2

(1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+

1

2

(1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1198612(119901+2)

1

2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

=

1

2

1205722minus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

= 119866 (120572)

(41)

where we have used 119872(119904) = int

119904

0119872(120591)d120591 ge 119898

0119904 ge 0 and

120572 = (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

(42)

It is easy to verify that119866(120572) is increasing in (0 120572lowast) decreasing

in (120572lowastinfin) and that 119866(120572) rarr minusinfin as 120572 rarr infin and

119866(120572)max = 119866 (120572lowast) =

1

2

1205722

lowastminus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

lowast= 1198641 (43)


where 120572lowastis given in (30) Since 119864(0) lt 119864

1 there exists 120572

3gt

120572lowastsuch that 119866(120572

3) = 119864(0) Set 120572

0= (1198971nabla11990602

2+ 1198972nablaV02

2)12

then from (41) we can get 119866(1205720) le 119864(0) = 119866(120572

3) which

implies that 1205720ge 1205723

Now we establish (40) by contradiction First we assumethat (40) is not true over [0 119879) then there exists 119905

0isin (0 119879)

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12

lt 1205723 (44)

By the continuity of 1198971nabla119906(119905)

2

2+ 1198972nablaV(119905)2

2we can choose 119905

0

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12


Again the use of (41) leads to

119864 (1199050) ge 119866 [(119897

1

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2)

12

]

gt 119866 (1205723) = 119864 (0)

(46)

This is impossible since 119864(119905) le 119864(0) for all 119905 isin [0 119879)Hence (40) is established

Lemma 9 (see [43 44]) Let 119871(119905) be a positive twice differen-tiable function which satisfies for 119905 gt 0 the inequality

119871 (119905) 11987110158401015840(119905) minus (1 + 120590) 119871

1015840(119905)2ge 0 (47)

for some 120590 gt 0 If 119871(0) gt 0 and 1198711015840(0) gt 0 then there exists a

time 119879lowastle 119871(0)[120590119871

1015840(0)] such that lim

119905rarr119879minus

lowast119871(119905) = infin

Proof of Theorem 5 Assume by contradiction that the solu-tion (119906 V) is global Then we consider 119871(119905) [0 119879] rarr R+

defined by

119871 (119905) = 119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591

+ (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) + 120573(119905 + 119879

0)2

(48)

where 119879 120573 and 1198790are positive constants to be chosen later

Then 119871(119905) gt 0 for all 119905 isin [0 119879] Furthermore

1198711015840(119905) = 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ nabla119906 (119905)2

2+ nablaV (119905)2

2minus (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 2120573 (119905 + 1198790)

= 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ 2int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+ 2int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591 + 2120573 (119905 + 119879

0)

(49)

and consequently

11987110158401015840(119905) = 2int

Ω

119906 (119905) 119906119905119905(119905) d119909 + 2int

Ω

V (119905) V119905119905(119905) d119909

+ 21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+ 2 (nabla119906 (119905) nabla119906

119905(119905))

+ 2 (nablaV (119905) nablaV119905(119905)) + 2120573

(50)

for almost every 119905 isin [0 119879] Testing the first equation ofsystem (1) with 119906 and the second equation of system (1) withV integrating the results over Ω using integration by partsand summing up we have

(119906119905119905(119905) 119906 (119905)) + (V

119905119905(119905) V (119905)) + (nabla119906 (119905) nabla119906

119905(119905))

+ (nablaV (119905) nablaV119905(119905))

= minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2minus119872(nablaV (119905)2

2) nablaV (119905)2

2

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

(51)

which implies

11987110158401015840(119905) = 2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus 2119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2

minus 2119872(nablaV (119905)22) nablaV (119905)2

2+ 2120573

minus 2int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus 2int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(52)

Therefore we have

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

minus119872(nablaV (119905)22) nablaV (119905)2

2+ 120573)


minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)

minus [119871 (119905) minus (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)

where Ψ(119905)119867(119905) [0 119879] rarr R+ are the functions defined by

Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0

[(nabla119906 (120591) nabla119906119905(120591)) + (nablaV (120591) nablaV

119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)

Using the Cauchy-Schwarz inequality we obtain

(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

(the similar inequality for V (119905) holds true)

(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

(the similar inequality for V (119905))

int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)


By Holderrsquos inequality and Youngrsquos inequality we obtain

120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

(the similar inequality for V (119905) holds true) (57)

The previous inequalities imply that 119867(119905) ge 0 for every 119905 isin

[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)

for almost every 119905 isin [0 119879] where Φ(119905) [0 119879] rarr R+ is themap defined by

Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)

For the fourth term on the right hand side of (59) we have

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω

nabla119906 (119904) sdot nabla119906 (119905) d119909 d119904

= int

119905

0

1198921(119905 minus 119904) int

Ω

nabla119906 (119905) sdot (nabla119906 (119904) minus nabla119906 (119905)) d119909 d119904

+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω

nablaV (119905) sdot (nablaV (119904) minus nablaV (119905)) d119909 d119904

+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)

Combining (59) (60) with (61) we get

Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω

nabla119906 (119905) (nabla119906 (119904) minus nabla119906 (119905)) d119909 d119904

minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω

nablaV (119905) (nablaV (119904) minus nablaV (119905)) d119909 d119904

minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2

= minus120576 (119892119894 nabla119906) (119905) minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)


for any 120576 gt 0 inserting (63) into (62) and utilizing (27) wehave

Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)

Using (29) and (A4) we have

Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)

times (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]

times nablaV (119905)22

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)

If 120575 lt 0 that is 119864(0) lt 0 we choose 120576 = 2(119901 + 2) in (65)and 120573 small enough such that 120573 le minus2119864(0) Then by (32) wehave

Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)

If 0 le 120575 lt 1 that is 0 le 119864(0) lt 1205751198641lt 1198641 we choose

120576 = 2(119901 + 2)(1 minus 120575) + 2120575 and 120573 = 2(1205751198641minus 119864(0)) in (65) Then

we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)

By (32) we have (notice that 120575 = 120575 due to 120575 ge 0)

int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)

for 119894 = 1 2 Therefore from the above two inequalities and(A2) we can get

Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12


by Lemma 8 there exists a constant 1205723gt 120572lowastsuch that

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)

which implies119901 + 1

2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)

It follows from (67) and (73) that

Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)

Therefore by (58) (66) and (74) we obtain

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)

for almost every 119905 isin [0 119879] By (49) we then choose 1198790

sufficiently large such that

(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)

Then by (76) and (77) we choose 119879 large enough so that

119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)

which ensures that 119879 gt (2(119901+1))(119871(0)1198711015840(0)) As (119901+3)2 gt

1 letting 120590 = (119901 + 1)2 we can select 119879lowastsuch that 119879

lowastle

119871(0)1205901198711015840(0) le 119879 By using Lemma 9 we get lim


lowast119871(119905) =

infin This implies that

lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)

which is a contradiction Thus 119879 lt infin

4 Blow-Up of Solutions withArbitrarily Positive Initial Energy

In this section we prove the second blow-up result(Theorem 6) for solutions with arbitrarily positive initialenergy In order to attain our aim we need the following threelemmas

Lemma 10 (see [4]) Let 120575lowast gt 0 and 119861(119905) isin 1198622(0infin) be a

nonnegative function satisfying

11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)

for 119905 gt 0 where 1198700is a constant and 119903

2= 2(120575

lowast+ 1) minus

2radic(120575lowast+ 1)120575lowast is the smallest root of the equation

1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)


Lemma 11 (see [4]) If ℎ(t) is a nonincreasing function on[0infin) and satisfies the differential inequality

ℎ1015840(119905)2ge 119886lowast+ 119887lowastℎ(119905)2+1120575

lowast

for 119905 ge 0 (84)

where 119886lowastgt 0 120575

lowastgt 0 and 119887

lowastgt 0 then there exists a finite

time 119879lowast such that lim119905rarr119879

lowastminusℎ(119905) = 0 and the upper bound of119879lowast is estimated by

119879lowastle 2(3120575lowast+1)2120575

lowast 120575lowast119888

radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)

where 119888 = (119886lowast119887lowast)2+1120575

lowast

For the next lemma we define

119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)

Lemma 12 Assume that the conditions ofTheorem 6 hold andlet (119906 V) be a solution of (1) then

1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)

Proof By (86) we have

1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω

nabla119906 sdot nabla119906119905d119909

+ 2int

Ω

nablaV sdot nablaV119905d119909

(89)

Testing the first equation of system (1) with 119906 and testing thesecond equation of system (1) with V and plugging the resultsinto the expression of11987010158401015840(119905) we obtain

11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2

minus 2119872(nablaV22) nablaV2

2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω

1198921(119905 minus 119904) nabla119906 (119904) sdot nabla119906 (119905) d119909 d119904

+ 2int

119905

0

int

Ω

1198922(119905 minus 119904) nablaV (119904) sdot nablaV (119905) d119909 d119904

(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)

By using Holderrsquos inequality and Youngrsquos inequality we have

2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)


Then taking (92) and (93) into account we obtain

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)

minus [2119872(nablaV22)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)

Thus by (A4) and (33) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω

nabla119906 sdot nabla119906119905d119909 d120591

nablaV (119905)22minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω

nablaV sdot nablaV119905d119909 d120591

(96)

ByHolderrsquos inequality and Youngrsquos inequality we obtain from(96)

nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)

ByHolderrsquos inequality and Youngrsquos inequality again it followsfrom (86) (88) and (97) that

1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)

In view of (95) and (98) we have

11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)

Then 119861(119905) satisfies (80) for 120575lowast = (119901+1)2 By (81) we see thatif

1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)

which is satisfied by the second hypothesis to Theorem 6thenwe get fromLemma 10 that 119870

1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt

0 Thus the proof of Lemma 12 is completed

In what follows we find an estimate for the life spanof 119870(119905) and proveTheorem 6

Proof of Theorem 6 Let

ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)

where 120575lowast = (119901 + 1)2 and 1198791gt 0 is a certain constant which

will be specified later Then we have

ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus (1 + 120575lowast) (1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)


For simplicity we denote

119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)

It follows from (88) (96) Holderrsquos inequality and Youngrsquosinequality that

1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)

Applying (107)ndash(110) we obtain

119881 (119905) ge [(minus4 minus 8120575lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus 4 (1 + 120575lowast) (radic119875119877 + radic119876119878)

2

(111)

From (104) and (98) we deduce that

119881 (119905) ge (minus4 minus 8120575lowast) 119864 (0) ℎ(119905)

minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 4 (1 + 120575lowast) [(119877 + 119878) (119875 + 119876) minus (radic119875119877 + radic119876119878)

2

]

(112)

By Cauchy-Schwarz inequality the last term in the aboveinequality is nonnegative Hence we have


minus1120575lowast

119905 ge 0 (113)

Therefore by (106) and (113) we get

ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)

Note that by Lemma 12 ℎ1015840(119905) lt 0 for 119905 gt 0 Multiplying (114)by ℎ1015840(119905) and integrating from 0 to 119905 we obtain

ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

minus8119864 (0) ℎ(0)minus1120575lowast

]

(116)

1198871= 8120575lowast2119864 (0) (117)

We observe that 1198861gt 0 if

119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)

Then by Lemma 11 there exists a finite time 119879lowast such that

lim119905rarr119879

lowastminusℎ(119905) = 0 Moreover the upper bounds of 119879lowast areestimated by

119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)

This completes the proof

Remark 13 The choice of1198791in (104) is possible provided that

1198791ge 119879lowast

5 General Decay of Solutions

In this section we prove the general decay of solutions ofsystem (1) The method of proof is similar to that of [42Theorem 35] We first state a lemma which is similar to theone first proved by Vitillaro in [33] to study a class of a scalarwave equation

Lemma 14 ([23 Lemma 32]) Suppose that (20) (A1) and(1198602) hold Let (119906 V) be the solution of system (1) Assumefurther that 119864(0) lt 119864

1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)


The auxiliary functionals 119868(119905) 119869(119905) of problem (1) aredefined as

119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)

Lemma 15 Suppose that (20) (A1) and (1198602) hold Let (119906 V)be the solution of system (1) Assume further that 119864(0) lt 119864

1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)

Proof Since 119864(0) lt 1198641and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)

it follows from Lemma 14 and (30) that

1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

times [1 minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)

for 119905 isin [0 119879) where we have used (24) Further by (123) wehave

119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)

times [(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)

We note that the following functional was introduced in[42]

1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)

where 1205761and 1205762are some positive constants and

120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)

Here we use the same functional (132) but choose 120585119894(119905) equiv

1 (119894 = 1 2) in (133) and (134)

Lemma 16 There exist two positive constants 1205731and 120573

2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)

Proof From Holderrsquos inequality Youngrsquos inequalityLemma 2 (36) and (125) we deduce1003816100381610038161003816120601 (119905)

1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)

Applying Holderrsquos inequality Youngrsquos inequality andLemma 2 again it follows that

int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)

Similarly applying Holderrsquos inequality Youngrsquos inequalityand Lemma 2 again we have

int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)

It follows from (137) (138) (36) and (126) that

1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)

If we take 1205981and 1205982to be sufficiently small then (135) follows

from (132) (136) and (139)


Lemma 17 Under the conditions ofTheorem 7 the functional120601(119905) defined by (133) (with 120585

119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2

minus [119897 minus 120598 (1198980minus 119897 +

1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909

minus 1198981(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2) forall120598 gt 0

(140)

Proof By using the equations in (1) and setting 120585119894(119905) equiv 1 (119894 =

1 2) in (133) we easily see that

1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

minus119872(nabla1199062

2) nabla119906

2

2minus119872(nablaV2

2) nablaV2

2

+ int

Ω

nabla119906 (119905) sdot int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω

nablaV (119905) sdot int119905

0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω

nabla119906119905sdot nabla119906d119909 minus int

Ω

nablaV119905sdot nablaVd119909 + 2 (119901 + 2)

times int

Ω

119865 (119906 V) d119909

(141)

By Cauchy-Schwarz inequality and Youngrsquos inequality wehave

int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|

times |nabla119906 (119904) minus nabla119906 (119905)| d119904 d119909 le (1 + 120598) (int

119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)

In the same way we can get

int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω

nabla119906119905sdot nabla119906d119909 le

120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω

nablaV119905sdot nablaVd119909 le

120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)

Using (141)ndash(145) we obtain

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2

minus 1198980nablaV22minus 1198981(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)

So Lemma 17 is established


119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)

Proof By using the equations in (1) and set 120585119894(119905) equiv 1 (119894 =

1 2) in (134) we observe that

1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0

1198921(119905 minus 119904) (nabla119906 (119905) minus nabla119906 (119904)) d119904 d119909

+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0

1198922(119905 minus 119904) (nablaV (119905) minus nablaV (119904)) d119904 d119909

minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0

1198921(119905 minus 119904) (nabla119906 (119905) minus nabla119906 (119904)) d119904] d119909

minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0

1198922(119905 minus 119904) (nablaV (119905) minus nablaV (119904)) d119904] d119909

+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0

1198922(119905 minus 119904) (nablaV (119905) minus nablaV (119904)) d119904 d119909]

minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)

In what follows we will estimate 1198681 119868

7in (149) By Youngrsquos

inequality (A2) and (125) we get

10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)

For 1198683in (149) applying (A2)Holderrsquos inequality andYoungrsquos

inequality we deduce

10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0

1198921(119905 minus 119904) (nabla119906 (119905) minus nabla119906 (119904)) d119904)

2

d119909

le 120578int

Ω

[int

119905

0

1198921(119905 minus 119904) (|nabla119906 (119904) minus nabla119906 (119905)| + |nabla119906 (119905)|) d119904]

2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)

In the same way we have

10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)

By Youngrsquos inequality Lemma 2 and (125) we see that

10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)

By Youngrsquos inequality and Lemma 2 again we obtain

10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)

Combining (149)ndash(157) we complete the proof of Lemma 18

Proof of Theorem 7 Since119892119894are positive we have for any 119905

0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)

By using (28) (132) (140) and (147) a series of computationsfor 119905 ge 119905

0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]

minus [11989811205761minus 11989811205762120578 (1198980minus 119897)]

times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus (12057621198923minus 1205762120578 minus 1205761) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)

We choose 1205761 1205762 and 120578 small enough such that

1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)

and 120598 = 1205761 then we can check that

1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576

2so small that (162) remains valid and

further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)

which yields (if needed one can choose 1205762sufficiently small)

1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)

where 119888lowastis some positive constant It follows from (165) (A3)

and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) minus 119862119864

1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840

lowast(119905) le minus119888

lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866

1(119905) + 119862119864(119905) is equivalent to 119864(119905) due

to (135) and 119896 is a positive constant A simple integration of(167) leads to

119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to the anonymous referees and theeditors for their useful remarks and comments on an earlyversion of this work This work was partly supported bythe National Natural Science Foundation of China (Grantno 11301277) the Qing Lan Project of Jiangsu Provinceand the Chinese Ministry of Finance Project (Grant noGYHY200906006)

References

[1] R Torrejon and J M Yong ldquoOn a quasilinear wave equationwith memoryrdquo Nonlinear Analysis Theory Methods amp Applica-tions vol 16 no 1 pp 61ndash78 1991

[2] J E Munoz Rivera ldquoGlobal solution on a quasilinear waveequation with memoryrdquo Unione Matematica Italiana B vol 8no 2 pp 289ndash303 1994

[3] S-T Wu and L-Y Tsai ldquoOn global existence and blow-upof solutions for an integro-differential equation with strongdampingrdquo Taiwanese Journal of Mathematics vol 10 no 4 pp979ndash1014 2006

[4] M-R Li and L-Y Tsai ldquoExistence and nonexistence of globalsolutions of some system of semilinear wave equationsrdquoNonlin-ear Analysis Theory Methods amp Applications vol 54 no 8 pp1397ndash1415 2003

[5] S-TWu ldquoExponential energy decay of solutions for an integro-differential equation with strong dampingrdquo Journal of Mathe-matical Analysis and Applications vol 364 no 2 pp 609ndash6172010

[6] T Matsuyama and R Ikehata ldquoOn global solutions and energydecay for the wave equations of Kirchhoff type with nonlineardamping termsrdquo Journal of Mathematical Analysis and Applica-tions vol 204 no 3 pp 729ndash753 1996

[7] K Ono ldquoBlowing up and global existence of solutions for somedegenerate nonlinear wave equations with some dissipationrdquoNonlinear Analysis Theory Methods amp Applications vol 30 no7 pp 4449ndash4457 1997

[8] K Ono ldquoGlobal existence decay and blowup of solutions forsome mildly degenerate nonlinear Kirchhoff stringsrdquo Journal ofDifferential Equations vol 137 no 2 pp 273ndash301 1997

[9] K Ono ldquoOn global existence asymptotic stability and blowingup of solutions for some degenerate non-linear wave equationsof Kirchhoff type with a strong dissipationrdquo MathematicalMethods in the Applied Sciences vol 20 no 2 pp 151ndash177 1997

[10] K Ono ldquoOn global solutions and blow-up solutions of non-linear Kirchhoff strings with nonlinear dissipationrdquo Journal of

Mathematical Analysis and Applications vol 216 no 1 pp 321ndash342 1997

[11] J Y Park and J J Bae ldquoOn existence of solutions of nondegen-erate wave equations with nonlinear damping termsrdquo NihonkaiMathematical Journal vol 9 no 1 pp 27ndash46 1998

[12] A Benaissa and S A Messaoudi ldquoBlow-up of solutions forthe Kirchhoff equation of 119902-Laplacian type with nonlineardissipationrdquo Colloquium Mathematicum vol 94 no 1 pp 103ndash109 2002

[13] S-T Wu and L-Y Tsai ldquoOn a system of nonlinear waveequations of Kirchhoff type with a strong dissipationrdquo TamkangJournal of Mathematics vol 38 no 1 pp 1ndash20 2007

[14] MM Cavalcanti V N Domingos Cavalcanti and J A SorianoldquoExponential decay for the solution of semilinear viscoelasticwave equations with localized dampingrdquo Electronic Journal ofDifferential Equations vol 2002 no 44 14 pages 2002

[15] E Zuazua ldquoExponential decay for the semilinear wave equationwith locally distributed dampingrdquo Communications in PartialDifferential Equations vol 15 no 2 pp 205ndash235 1990

[16] M M Cavalcanti and H P Oquendo ldquoFrictional versus vis-coelastic damping in a semilinear wave equationrdquo SIAM Journalon Control and Optimization vol 42 no 4 pp 1310ndash1324 2003

[17] S A Messaoudi ldquoBlow up and global existence in a nonlinearviscoelastic wave equationrdquo Mathematische Nachrichten vol260 pp 58ndash66 2003

[18] S A Messaoudi ldquoBlow-up of positive-initial-energy solutionsof a nonlinear viscoelastic hyperbolic equationrdquo Journal ofMathematical Analysis andApplications vol 320 no 2 pp 902ndash915 2006

[19] W Liu ldquoGlobal existence asymptotic behavior and blow-up ofsolutions for a viscoelastic equation with strong damping andnonlinear sourcerdquo Topological Methods in Nonlinear Analysisvol 36 no 1 pp 153ndash178 2010

[20] X Han and M Wang ldquoGlobal existence and blow-up ofsolutions for a system of nonlinear viscoelastic wave equationswith damping and sourcerdquoNonlinear Analysis Theory Methodsamp Applications vol 71 no 11 pp 5427ndash5450 2009

[21] S A Messaoudi and B Said-Houari ldquoGlobal nonexistenceof positive initial-energy solutions of a system of nonlinearviscoelastic wave equations with damping and source termsrdquoJournal of Mathematical Analysis and Applications vol 365 no1 pp 277ndash287 2010

[22] F Liang and H Gao ldquoExponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic waveequations with strong dampingrdquo Boundary Value Problems vol2011 article 22 19 pages 2011

[23] G Li L Hong and W Liu ldquoExponential energy decay of solu-tions for a system of viscoelastic wave equations of Kirchhofftype with strong dampingrdquo Applicable Analysis vol 92 no 5pp 1046ndash1062 2013

[24] D Andrade and L H Fatori ldquoThe nonlinear transmissionproblem with memoryrdquo Boletim da Sociedade Paranaense deMatematica vol 22 no 1 pp 106ndash118 2004

[25] M M Cavalcanti V N Domingos Cavalcanti J S PratesFilho and J A Soriano ldquoExistence and exponential decay for aKirchhoff-Carrier model with viscosityrdquo Journal of Mathemati-cal Analysis and Applications vol 226 no 1 pp 40ndash60 1998

[26] M M Cavalcanti V N Domingos Cavalcanti and M LSantos ldquoExistence and uniform decay rates of solutions to adegenerate system with memory conditions at the boundaryrdquoApplied Mathematics and Computation vol 150 no 2 pp 439ndash465 2004


[27] V Komornik Exact Controllability and Stabilization Researchin Applied Mathematics Masson Paris France 1994

[28] W Liu ldquoGeneral decay rate estimate for a viscoelastic equationwith weakly nonlinear time-dependent dissipation and sourcetermsrdquo Journal of Mathematical Physics vol 50 no 11 ArticleID 113506 17 pages 2009

[29] J Y Park and J J Bae ldquoVariational inequality for quasilinearwave equations with nonlinear damping termsrdquo NonlinearAnalysis Theory Methods amp Applications vol 50 no 8 pp1065ndash1083 2002

[30] W Liu ldquoGeneral decay and blow-up of solution for a quasilinearviscoelastic problemwith nonlinear sourcerdquoNonlinearAnalysisTheory Methods amp Applications vol 73 no 6 pp 1890ndash19042010

[31] W Liu and J Yu ldquoOn decay and blow-up of the solution for aviscoelastic wave equation with boundary damping and sourcetermsrdquoNonlinear AnalysisTheory Methods amp Applications vol74 no 6 pp 2175ndash2190 2011

[32] B Said-Houari ldquoGlobal nonexistence of positive initial-energysolutions of a system of nonlinear wave equations with dampingand source termsrdquo Differential and Integral Equations vol 23no 1-2 pp 79ndash92 2010

[33] E Vitillaro ldquoGlobal nonexistence theorems for a class of evolu-tion equations with dissipationrdquo Archive for Rational Mechanicsand Analysis vol 149 no 2 pp 155ndash182 1999

[34] S-T Wu ldquoBlow-up of solutions for a system of nonlinearwave equations with nonlinear dampingrdquo Electronic Journal ofDifferential Equations vol 2009 no 105 11 pages 2009

[35] K Agre and M A Rammaha ldquoSystems of nonlinear waveequations with damping and source termsrdquo Differential andIntegral Equations vol 19 no 11 pp 1235ndash1270 2006

[36] M-R Li and L-Y Tsai ldquoOn a system of nonlinear waveequationsrdquo Taiwanese Journal of Mathematics vol 7 no 4 pp557ndash573 2003

[37] W Liu ldquoUniform decay of solutions for a quasilinear system ofviscoelastic equationsrdquo Nonlinear Analysis Theory Methods ampApplications vol 71 no 5-6 pp 2257ndash2267 2009

[38] F Sun and M Wang ldquoGlobal and blow-up solutions fora system of nonlinear hyperbolic equations with dissipativetermsrdquoNonlinear AnalysisTheory Methods amp Applications vol64 no 4 pp 739ndash761 2006

[39] S-T Wu ldquoOn decay and blow-up of solutions for a system ofnonlinear wave equationsrdquo Journal of Mathematical Analysisand Applications vol 394 no 1 pp 360ndash377 2012

[40] S Yu ldquoPolynomial stability of solutions for a system of non-linear viscoelastic equationsrdquo Applicable Analysis vol 88 no 7pp 1039ndash1051 2009

[41] R A Adams Sobolev Spaces vol 65 of Pure and AppliedMathematics Academic Press New York NY USA 1975

[42] S A Messaoudi ldquoGeneral decay of the solution energy ina viscoelastic equation with a nonlinear sourcerdquo NonlinearAnalysis Theory Methods amp Applications vol 69 no 8 pp2589ndash2598 2008

[43] H A Levine ldquoSome nonexistence and instability theorems forsolutions of formally parabolic equations of the form 119875119906

119905119905=

minus119860119906 +F(119906)rdquo Archive for Rational Mechanics and Analysis vol51 pp 371ndash386 1973

[44] H A Levine ldquoInstability and nonexistence of global solutionsto nonlinear wave equations of the form 119875119906

119905119905= minus119860119906 + F(119906)rdquo

Transactions of the American Mathematical Society vol 192 pp1ndash21 1974

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


in [5] under a weaker condition on 119892 (ie 1198921015840(119905) le 0 for119905 ge 0) For a single wave equation of Kirchhoff type withoutthe viscoelastic term we refer the reader to Matsuyama andIkehata [6] and Ono [7ndash10]

Many results concerning local existence global existencedecay and blow-up of solutions for a system of waveequations of Kirchhoff type without viscoelastic terms (ie119892119894

= 0 119894 = 1 2) have also been extensively studiedFor example Park and Bae [11] considered the system ofwave equations with nonlinear dampings for 119891

1(119906 V) =

120583|119906|119902minus1

119906 and 1198912(119906 V) = 120583|V|119902minus1V 119902 ge 1 and showed the

global existence and asymptotic behavior of solutions undersome restrictions on the initial energy Later Benaissa andMessaoudi [12] discussed blow-up properties for negativeinitial energy Recently Wu and Tsai [13] studied the system(1) for 119892

119894= 0 (119894 = 1 2) Under some suitable assumptions

on 119891119894(119894 = 1 2) they proved local existence of solutions by

applying the Banach fixed point theorem and the blow-up ofsolutions by using the method of Li and Tsai in [4] wherethree different cases on the sign of the initial energy 119864(0) areconsidered

In the case of 119872 equiv 1 and in the presence of viscoelasticterm (ie 119892 = 0) Cavalcanti et al [14] studied the equationthat was subject to a locally distributed dissipation

119906119905119905minus Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 + 119886 (119909) 119906119905+ |119906|120574119906 = 0


(3)

with the same initial and boundary conditions as that of (2)and proved an exponential decay rate This work extendedthe result of Zuazua [15] in which he considered (3) with119892 = 0 and the localized linear damping By using thepiecewise multipliers method Cavalcanti and Oquendo [16]investigated the equation

119906119905119905minus 1198960Δ119906 + int

119905

0

div [119886 (119909) 119892 (119905 minus 119904) nabla119906 (119904)] d119904

+ 119887 (119909) ℎ (119906119905) + 119891 (119906) = 0 (119909 119905) isin Ω times (0infin)

(4)

with the same initial and boundary conditions as that of (2)Under the similar conditions on the relaxation function 119892 asabove and 119886(119909) + 119887(119909) ge 120575 gt 0 for all 119909 isin Ω they improvedthe results of [14] by establishing exponential stability forexponential decay function 119892 and linear function ℎ andpolynomial stability for polynomial decay function 119892 andnonlinear function ℎ respectively

Concerning blow-up results Messaoudi [17] consideredthe equation

119906119905119905minus Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 + 119886119906119905

1003816100381610038161003816119906119905

1003816100381610038161003816

119898minus2

= 119887|119906|119903minus2

119906


(5)

He proved that any weak solution with negative initial energyblows up in finite time if 119903 gt 119898 and

int

infin

0

119892 (119904) d119904 le 119903 minus 2

119903 minus 2 + 1119903

(6)

while exists globally for any initial data in the appropriatespace if 119898 ge 119903 This result was improved by the same authorin [18] for positive initial energy under suitable conditions on119892 119898 and 119903 Recently Liu [19] studied the equation

119906119905119905minus Δ119906 + int

119905

0

119892 (119905 minus 119904) Δ119906 (119904) d119904 minus 120596Δ119906119905+ 120583119906119905= |119906|119903minus2

119906


(7)

with the same initial and boundary condition as that of (2)By virtue of convexity technique and supposing that

int

infin

0

119892 (119904) d119904 le 119903 minus 2

119903 minus 2 + 1 [(1 minus120575)

2

119903 + 2120575 (1 minus120575)]

(8)

where 120575 = max0 120575 he proved that the solution with

nonpositive initial energy as well as positive initial energyblows up in finite time

We should mention that the following system

119906119905119905minus Δ119906 + int

119905

0

1198921(119905 minus 120591) Δ119906 (120591) d120591 + 1003816

100381610038161003816119906119905

1003816100381610038161003816

119898minus1

119906119905= 1198911(119906 V)

(119909 119905) isin Ω times (0 119879)


119905

0

1198922(119905 minus 120591) ΔV (120591) d120591 + 1003816

100381610038161003816V119905

1003816100381610038161003816

120574minus1V119905= 1198912(119906 V)

(119909 119905) isin Ω times (0 119879)

119906 (119909 119905) = 0 V (119909 119905) = 0

(119909 119905) isin 120597Ω times [0 119879)

119906 (119909 0) = 1199060(119909) 119906

119905(119909 0) = 119906

1(119909) 119909 isin Ω

V (119909 0) = V0(119909) V

119905(119909 0) = V

1(119909) 119909 isin Ω

(9)

was considered by Han and Wang in [20] where Ω is abounded domain with smooth boundary 120597Ω in R119899 119899 =

1 2 3Under suitable assumptions on the functions119892119894 119891119894(119894 =

1 2) the initial data and the parameters in the above problemestablished local existence global existence and blow-upproperty (the initial energy 119864(0) lt 0) This latter blow-upresult has been improved byMessaoudi and Said-Houari [21]into certain solutions with positive initial energy RecentlyLiang and Gao in [22] investigated the following problem

119906119905119905minus Δ119906 + int

119905

0

1198921(119905 minus 120591)Δ119906 (120591) d120591 minus Δ119906

119905= 1198911(119906 V)

(119909 119905) isin Ω times (0 119879)


119905

0

1198922(119905 minus 120591)ΔV (120591) d120591 minus ΔV

119905= 1198912(119906 V)

(119909 119905) isin Ω times (0 119879)

(10)

with the same initial and boundary conditions as that of (9)Under suitable assumptions on the functions 119892

119894 119891119894(119894 = 1 2)








119894(119905) (119894 = 1 2)





119872(119904) ge 1198980gt 0 (11)



1198980minus int

infin

0

119892119894(119904) d119904 = 119897

119894gt 0 (12)


and 1205852(119905) such that

1198921015840

119894(119905) le minus120585

119894(119905) 119892119894(119905) (119894 = 1 2) for 119905 ge 0 (13)

and 120585119894(119905) satisfies

1205851015840

119894(119905) le 0 forall119905 gt 0 int

+infin

0

120585119894(119905) d119905 = infin (119894 = 1 2)

(14)



where 119872(119904) = int

119904

0119872(120591)d120591





2 (119901 + 2)119872 (119904) minus 2119872 (119904) 119904

= 2 (119901 + 2) (1198980119904 +

1198981

120574 + 1

119904120574+1

) minus 21198980119904 minus 2119898

1119904120574+1

= 2 (119901 + 1)1198980119904 +

2 (119901 + 1 minus 120574)

120574 + 1

1198981119904120574+1

ge 2 (119901 + 1)1198980119904

(16)


(119892119894 nabla119908) (119905) = int

119905

0


2

2d119904

119897 = min 1198971 1198972

1198920(119905) = max 119892

1(119905) 1198922(119905) forall119905 ge 0

sdot119902= sdot119871119902(Ω)


(17)


(119906 V) = int

Ω

119906 (119909) V (119909) d119909 (18)


2(119906 V) (see also [21])

1198911(119906 V) = [119886|119906 + V|2(119901+1) (119906 + V) + 119887|119906|

119901119906|V|119901+2]

1198912(119906 V) = [119886|119906 + V|2(119901+1) (119906 + V) + 119887|119906|

119901+2|V|119901V]

(19)


minus 1 lt 119901 if 119899 = 1 2


(3 minus 119899)

(119899 minus 2)

if 119899 ge 3

(20)


1199061198911(119906 V) + V119891

2(119906 V) = 2 (119901 + 2) 119865 (119906 V) forall (119906 V) isin R

2

(21)

where

119865 (119906 V) =1

2 (119901 + 2)

[119886|119906 + V|2(119901+2) + 2119887|119906V|119901+2] (22)




2119899(119899 minus 2) then

119906120588le 119862nabla119906

2for 119906 isin 119867

1

0(Ω) (23)




1

0(Ω) cap

1198672(Ω)) times (119867

1

0(Ω) cap 119867

2(Ω)) one has

119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

le 1205781(1198971nabla1199062

2+ 1198972nablaV22)

119901+2

(24)



0 V0

isin 1198671

0(Ω) cap 119867

2(Ω) and 119906

1 V1



119906 V isin 119862 ([0 119879) 1198671

0(Ω) cap 119867

2(Ω))

119906119905 V119905isin 119862 ([0 119879) 119871

2(Ω)) cap 119871

2([0 119879) 119867

1

0(Ω))

(25)


(1) 119879 = infin

(2) lim119905rarr119879

minus

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ Δ119906

2

2+ ΔV2

2) = infin

(26)


119864 (119905) =

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1

2

(int

119905

0

1198921(119904) d119904) nabla119906

2

2

minus

1

2

(int

119905

0

1198922(119904) d119904) nablaV2

2minus int

Ω

119865 (119906 V) d119909

+

1

2

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2

119872(nabla1199062

2) +

1

2

119872(nablaV22) for 119905 ge 0

(27)


1198641015840(119905)

= minus (1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2) minus

1

2

[1198921(119905) nabla119906

2

2+ 1198922(119905) nablaV2

2]

+

1

2

[(1198921015840

1 nabla119906) (119905) + (119892

1015840


(28)

Then we have

119864 (119905) = 119864 (0) minus int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

minus

1

2

int

119905

0

[1198921(120591) nabla119906 (120591)

2

2+ 1198922(120591) nablaV (120591)2

2] d120591

+

1

2

int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

(29)

We introduce

1198611= 12057812(119901+2)

1 120572

lowast= 119861minus(119901+2)(119901+1)

1

1198641= (

1

2

minus

1

2 (119901 + 2)

)1205722

lowast

(30)




0 V0) (1199061 V1) satisfy

119864 (0) lt 1205751198641 (119897

1

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus120575)

2

(119901 + 2) +120575 (1 minus

120575)]

(119894 = 1 2)

(32)




int

infin

0

119892119894(119904) d119904 le

2 (119901 + 1)1198980

2 (119901 + 1) + 12 (119901 + 2)

(119894 = 1 2) (33)



0(Ω)cap119867

2(Ω) and 119906

1 V1isin


0 lt 119864 (0)

lt min

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(34)


119879lowastle 2(3119901+5)2(119901+1)

(119901 + 1) 119888

2radic1198861

1 minus [1 + 119888ℎ (0)]minus1(119901+1)

(35)

where ℎ(0) = [11990602

2+ V02

2+ 1198791(nabla11990602

2+ nablaV

02

2)]minus(119901+1)2



0+ 1198981119904120574 1198980gt 0 119898

1ge 0


119864 (119905) = 119864 (119906 (119905) V (119905))

=

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(36)


0 1199061) isin (119867

1

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and (V

0 V1) isin

(1198671

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and satisfy 119864(0) lt 119864

1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12




119864 (119905) le 119870119890minus119896int119905

1199050120585(119904)d119904

119905 ge 1199050 (38)

where 120585(119905) = min1205851(119905) 1205852(119905)





(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 for 119905 isin [0 119879) (40)


119864 (119905) ge

1

2

(1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+

1

2

(1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1198612(119901+2)

1

2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

=

1

2

1205722minus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

= 119866 (120572)

(41)


119904

0119872(120591)d120591 ge 119898

0119904 ge 0 and

120572 = (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

(42)



119866(120572)max = 119866 (120572lowast) =

1

2

1205722

lowastminus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

lowast= 1198641 (43)




3gt


3) = 119864(0) Set 120572

0= (1198971nabla11990602

2+ 1198972nablaV02

2)12


3) which



0isin (0 119879)

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12

lt 1205723 (44)


2

2+ 1198972nablaV(119905)2


0

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12



119864 (1199050) ge 119866 [(119897

1

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2)

12

]

gt 119866 (1205723) = 119864 (0)

(46)



119871 (119905) 11987110158401015840(119905) minus (1 + 120590) 119871

1015840(119905)2ge 0 (47)


time 119879lowastle 119871(0)[120590119871



lowast119871(119905) = infin


defined by

119871 (119905) = 119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591

+ (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) + 120573(119905 + 119879

0)2

(48)



1198711015840(119905) = 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ nabla119906 (119905)2

2+ nablaV (119905)2

2minus (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 2120573 (119905 + 1198790)

= 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ 2int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+ 2int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591 + 2120573 (119905 + 119879

0)

(49)

and consequently

11987110158401015840(119905) = 2int

Ω

119906 (119905) 119906119905119905(119905) d119909 + 2int

Ω

V (119905) V119905119905(119905) d119909

+ 21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+ 2 (nabla119906 (119905) nabla119906

119905(119905))

+ 2 (nablaV (119905) nablaV119905(119905)) + 2120573

(50)


(119906119905119905(119905) 119906 (119905)) + (V

119905119905(119905) V (119905)) + (nabla119906 (119905) nabla119906

119905(119905))

+ (nablaV (119905) nablaV119905(119905))

= minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2minus119872(nablaV (119905)2

2) nablaV (119905)2

2

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

(51)

which implies

11987110158401015840(119905) = 2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus 2119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 2120573

minus 2int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus 2int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(52)

Therefore we have

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2+ 120573)


minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)


1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)


Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0


119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)


(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)



120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119894(119905) (119894 = 1 2)





119872(119904) ge 1198980gt 0 (11)



1198980minus int

infin

0

119892119894(119904) d119904 = 119897

119894gt 0 (12)


and 1205852(119905) such that

1198921015840

119894(119905) le minus120585

119894(119905) 119892119894(119905) (119894 = 1 2) for 119905 ge 0 (13)

and 120585119894(119905) satisfies

1205851015840

119894(119905) le 0 forall119905 gt 0 int

+infin

0

120585119894(119905) d119905 = infin (119894 = 1 2)

(14)



where 119872(119904) = int

119904

0119872(120591)d120591





2 (119901 + 2)119872 (119904) minus 2119872 (119904) 119904

= 2 (119901 + 2) (1198980119904 +

1198981

120574 + 1

119904120574+1

) minus 21198980119904 minus 2119898

1119904120574+1

= 2 (119901 + 1)1198980119904 +

2 (119901 + 1 minus 120574)

120574 + 1

1198981119904120574+1

ge 2 (119901 + 1)1198980119904

(16)


(119892119894 nabla119908) (119905) = int

119905

0


2

2d119904

119897 = min 1198971 1198972

1198920(119905) = max 119892

1(119905) 1198922(119905) forall119905 ge 0

sdot119902= sdot119871119902(Ω)


(17)


(119906 V) = int

Ω

119906 (119909) V (119909) d119909 (18)


2(119906 V) (see also [21])

1198911(119906 V) = [119886|119906 + V|2(119901+1) (119906 + V) + 119887|119906|

119901119906|V|119901+2]

1198912(119906 V) = [119886|119906 + V|2(119901+1) (119906 + V) + 119887|119906|

119901+2|V|119901V]

(19)


minus 1 lt 119901 if 119899 = 1 2


(3 minus 119899)

(119899 minus 2)

if 119899 ge 3

(20)


1199061198911(119906 V) + V119891

2(119906 V) = 2 (119901 + 2) 119865 (119906 V) forall (119906 V) isin R

2

(21)

where

119865 (119906 V) =1

2 (119901 + 2)

[119886|119906 + V|2(119901+2) + 2119887|119906V|119901+2] (22)




2119899(119899 minus 2) then

119906120588le 119862nabla119906

2for 119906 isin 119867

1

0(Ω) (23)




1

0(Ω) cap

1198672(Ω)) times (119867

1

0(Ω) cap 119867

2(Ω)) one has

119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

le 1205781(1198971nabla1199062

2+ 1198972nablaV22)

119901+2

(24)



0 V0

isin 1198671

0(Ω) cap 119867

2(Ω) and 119906

1 V1



119906 V isin 119862 ([0 119879) 1198671

0(Ω) cap 119867

2(Ω))

119906119905 V119905isin 119862 ([0 119879) 119871

2(Ω)) cap 119871

2([0 119879) 119867

1

0(Ω))

(25)


(1) 119879 = infin

(2) lim119905rarr119879

minus

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ Δ119906

2

2+ ΔV2

2) = infin

(26)


119864 (119905) =

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1

2

(int

119905

0

1198921(119904) d119904) nabla119906

2

2

minus

1

2

(int

119905

0

1198922(119904) d119904) nablaV2

2minus int

Ω

119865 (119906 V) d119909

+

1

2

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2

119872(nabla1199062

2) +

1

2

119872(nablaV22) for 119905 ge 0

(27)


1198641015840(119905)

= minus (1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2) minus

1

2

[1198921(119905) nabla119906

2

2+ 1198922(119905) nablaV2

2]

+

1

2

[(1198921015840

1 nabla119906) (119905) + (119892

1015840


(28)

Then we have

119864 (119905) = 119864 (0) minus int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

minus

1

2

int

119905

0

[1198921(120591) nabla119906 (120591)

2

2+ 1198922(120591) nablaV (120591)2

2] d120591

+

1

2

int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

(29)

We introduce

1198611= 12057812(119901+2)

1 120572

lowast= 119861minus(119901+2)(119901+1)

1

1198641= (

1

2

minus

1

2 (119901 + 2)

)1205722

lowast

(30)




0 V0) (1199061 V1) satisfy

119864 (0) lt 1205751198641 (119897

1

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus120575)

2

(119901 + 2) +120575 (1 minus

120575)]

(119894 = 1 2)

(32)




int

infin

0

119892119894(119904) d119904 le

2 (119901 + 1)1198980

2 (119901 + 1) + 12 (119901 + 2)

(119894 = 1 2) (33)



0(Ω)cap119867

2(Ω) and 119906

1 V1isin


0 lt 119864 (0)

lt min

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(34)


119879lowastle 2(3119901+5)2(119901+1)

(119901 + 1) 119888

2radic1198861

1 minus [1 + 119888ℎ (0)]minus1(119901+1)

(35)

where ℎ(0) = [11990602

2+ V02

2+ 1198791(nabla11990602

2+ nablaV

02

2)]minus(119901+1)2



0+ 1198981119904120574 1198980gt 0 119898

1ge 0


119864 (119905) = 119864 (119906 (119905) V (119905))

=

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(36)


0 1199061) isin (119867

1

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and (V

0 V1) isin

(1198671

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and satisfy 119864(0) lt 119864

1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12




119864 (119905) le 119870119890minus119896int119905

1199050120585(119904)d119904

119905 ge 1199050 (38)

where 120585(119905) = min1205851(119905) 1205852(119905)





(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 for 119905 isin [0 119879) (40)


119864 (119905) ge

1

2

(1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+

1

2

(1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1198612(119901+2)

1

2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

=

1

2

1205722minus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

= 119866 (120572)

(41)


119904

0119872(120591)d120591 ge 119898

0119904 ge 0 and

120572 = (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

(42)



119866(120572)max = 119866 (120572lowast) =

1

2

1205722

lowastminus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

lowast= 1198641 (43)




3gt


3) = 119864(0) Set 120572

0= (1198971nabla11990602

2+ 1198972nablaV02

2)12


3) which



0isin (0 119879)

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12

lt 1205723 (44)


2

2+ 1198972nablaV(119905)2


0

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12



119864 (1199050) ge 119866 [(119897

1

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2)

12

]

gt 119866 (1205723) = 119864 (0)

(46)



119871 (119905) 11987110158401015840(119905) minus (1 + 120590) 119871

1015840(119905)2ge 0 (47)


time 119879lowastle 119871(0)[120590119871



lowast119871(119905) = infin


defined by

119871 (119905) = 119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591

+ (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) + 120573(119905 + 119879

0)2

(48)



1198711015840(119905) = 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ nabla119906 (119905)2

2+ nablaV (119905)2

2minus (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 2120573 (119905 + 1198790)

= 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ 2int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+ 2int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591 + 2120573 (119905 + 119879

0)

(49)

and consequently

11987110158401015840(119905) = 2int

Ω

119906 (119905) 119906119905119905(119905) d119909 + 2int

Ω

V (119905) V119905119905(119905) d119909

+ 21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+ 2 (nabla119906 (119905) nabla119906

119905(119905))

+ 2 (nablaV (119905) nablaV119905(119905)) + 2120573

(50)


(119906119905119905(119905) 119906 (119905)) + (V

119905119905(119905) V (119905)) + (nabla119906 (119905) nabla119906

119905(119905))

+ (nablaV (119905) nablaV119905(119905))

= minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2minus119872(nablaV (119905)2

2) nablaV (119905)2

2

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

(51)

which implies

11987110158401015840(119905) = 2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus 2119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 2120573

minus 2int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus 2int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(52)

Therefore we have

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2+ 120573)


minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)


1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)


Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0


119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)


(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)



120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2119899(119899 minus 2) then

119906120588le 119862nabla119906

2for 119906 isin 119867

1

0(Ω) (23)




1

0(Ω) cap

1198672(Ω)) times (119867

1

0(Ω) cap 119867

2(Ω)) one has

119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

le 1205781(1198971nabla1199062

2+ 1198972nablaV22)

119901+2

(24)



0 V0

isin 1198671

0(Ω) cap 119867

2(Ω) and 119906

1 V1



119906 V isin 119862 ([0 119879) 1198671

0(Ω) cap 119867

2(Ω))

119906119905 V119905isin 119862 ([0 119879) 119871

2(Ω)) cap 119871

2([0 119879) 119867

1

0(Ω))

(25)


(1) 119879 = infin

(2) lim119905rarr119879

minus

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ Δ119906

2

2+ ΔV2

2) = infin

(26)


119864 (119905) =

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1

2

(int

119905

0

1198921(119904) d119904) nabla119906

2

2

minus

1

2

(int

119905

0

1198922(119904) d119904) nablaV2

2minus int

Ω

119865 (119906 V) d119909

+

1

2

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2

119872(nabla1199062

2) +

1

2

119872(nablaV22) for 119905 ge 0

(27)


1198641015840(119905)

= minus (1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2) minus

1

2

[1198921(119905) nabla119906

2

2+ 1198922(119905) nablaV2

2]

+

1

2

[(1198921015840

1 nabla119906) (119905) + (119892

1015840


(28)

Then we have

119864 (119905) = 119864 (0) minus int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

minus

1

2

int

119905

0

[1198921(120591) nabla119906 (120591)

2

2+ 1198922(120591) nablaV (120591)2

2] d120591

+

1

2

int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

(29)

We introduce

1198611= 12057812(119901+2)

1 120572

lowast= 119861minus(119901+2)(119901+1)

1

1198641= (

1

2

minus

1

2 (119901 + 2)

)1205722

lowast

(30)




0 V0) (1199061 V1) satisfy

119864 (0) lt 1205751198641 (119897

1

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus120575)

2

(119901 + 2) +120575 (1 minus

120575)]

(119894 = 1 2)

(32)




int

infin

0

119892119894(119904) d119904 le

2 (119901 + 1)1198980

2 (119901 + 1) + 12 (119901 + 2)

(119894 = 1 2) (33)



0(Ω)cap119867

2(Ω) and 119906

1 V1isin


0 lt 119864 (0)

lt min

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(34)


119879lowastle 2(3119901+5)2(119901+1)

(119901 + 1) 119888

2radic1198861

1 minus [1 + 119888ℎ (0)]minus1(119901+1)

(35)

where ℎ(0) = [11990602

2+ V02

2+ 1198791(nabla11990602

2+ nablaV

02

2)]minus(119901+1)2



0+ 1198981119904120574 1198980gt 0 119898

1ge 0


119864 (119905) = 119864 (119906 (119905) V (119905))

=

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(36)


0 1199061) isin (119867

1

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and (V

0 V1) isin

(1198671

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and satisfy 119864(0) lt 119864

1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12




119864 (119905) le 119870119890minus119896int119905

1199050120585(119904)d119904

119905 ge 1199050 (38)

where 120585(119905) = min1205851(119905) 1205852(119905)





(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 for 119905 isin [0 119879) (40)


119864 (119905) ge

1

2

(1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+

1

2

(1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1198612(119901+2)

1

2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

=

1

2

1205722minus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

= 119866 (120572)

(41)


119904

0119872(120591)d120591 ge 119898

0119904 ge 0 and

120572 = (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

(42)



119866(120572)max = 119866 (120572lowast) =

1

2

1205722

lowastminus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

lowast= 1198641 (43)




3gt


3) = 119864(0) Set 120572

0= (1198971nabla11990602

2+ 1198972nablaV02

2)12


3) which



0isin (0 119879)

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12

lt 1205723 (44)


2

2+ 1198972nablaV(119905)2


0

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12



119864 (1199050) ge 119866 [(119897

1

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2)

12

]

gt 119866 (1205723) = 119864 (0)

(46)



119871 (119905) 11987110158401015840(119905) minus (1 + 120590) 119871

1015840(119905)2ge 0 (47)


time 119879lowastle 119871(0)[120590119871



lowast119871(119905) = infin


defined by

119871 (119905) = 119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591

+ (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) + 120573(119905 + 119879

0)2

(48)



1198711015840(119905) = 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ nabla119906 (119905)2

2+ nablaV (119905)2

2minus (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 2120573 (119905 + 1198790)

= 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ 2int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+ 2int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591 + 2120573 (119905 + 119879

0)

(49)

and consequently

11987110158401015840(119905) = 2int

Ω

119906 (119905) 119906119905119905(119905) d119909 + 2int

Ω

V (119905) V119905119905(119905) d119909

+ 21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+ 2 (nabla119906 (119905) nabla119906

119905(119905))

+ 2 (nablaV (119905) nablaV119905(119905)) + 2120573

(50)


(119906119905119905(119905) 119906 (119905)) + (V

119905119905(119905) V (119905)) + (nabla119906 (119905) nabla119906

119905(119905))

+ (nablaV (119905) nablaV119905(119905))

= minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2minus119872(nablaV (119905)2

2) nablaV (119905)2

2

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

(51)

which implies

11987110158401015840(119905) = 2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus 2119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 2120573

minus 2int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus 2int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(52)

Therefore we have

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2+ 120573)


minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)


1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)


Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0


119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)


(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)



120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0(Ω)cap119867

2(Ω) and 119906

1 V1isin


0 lt 119864 (0)

lt min

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(34)


119879lowastle 2(3119901+5)2(119901+1)

(119901 + 1) 119888

2radic1198861

1 minus [1 + 119888ℎ (0)]minus1(119901+1)

(35)

where ℎ(0) = [11990602

2+ V02

2+ 1198791(nabla11990602

2+ nablaV

02

2)]minus(119901+1)2



0+ 1198981119904120574 1198980gt 0 119898

1ge 0


119864 (119905) = 119864 (119906 (119905) V (119905))

=

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(36)


0 1199061) isin (119867

1

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and (V

0 V1) isin

(1198671

0(Ω) cap 119867

2(Ω)) times 119871

2(Ω) and satisfy 119864(0) lt 119864

1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12




119864 (119905) le 119870119890minus119896int119905

1199050120585(119904)d119904

119905 ge 1199050 (38)

where 120585(119905) = min1205851(119905) 1205852(119905)





(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 for 119905 isin [0 119879) (40)


119864 (119905) ge

1

2

(1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+

1

2

(1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1

2 (119901 + 2)

(119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge

1

2

[1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2]

minus

1198612(119901+2)

1

2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

=

1

2

1205722minus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

= 119866 (120572)

(41)


119904

0119872(120591)d120591 ge 119898

0119904 ge 0 and

120572 = (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

(42)



119866(120572)max = 119866 (120572lowast) =

1

2

1205722

lowastminus

1198612(119901+2)

1

2 (119901 + 2)

1205722(119901+2)

lowast= 1198641 (43)




3gt


3) = 119864(0) Set 120572

0= (1198971nabla11990602

2+ 1198972nablaV02

2)12


3) which



0isin (0 119879)

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12

lt 1205723 (44)


2

2+ 1198972nablaV(119905)2


0

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12



119864 (1199050) ge 119866 [(119897

1

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2)

12

]

gt 119866 (1205723) = 119864 (0)

(46)



119871 (119905) 11987110158401015840(119905) minus (1 + 120590) 119871

1015840(119905)2ge 0 (47)


time 119879lowastle 119871(0)[120590119871



lowast119871(119905) = infin


defined by

119871 (119905) = 119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591

+ (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) + 120573(119905 + 119879

0)2

(48)



1198711015840(119905) = 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ nabla119906 (119905)2

2+ nablaV (119905)2

2minus (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 2120573 (119905 + 1198790)

= 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ 2int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+ 2int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591 + 2120573 (119905 + 119879

0)

(49)

and consequently

11987110158401015840(119905) = 2int

Ω

119906 (119905) 119906119905119905(119905) d119909 + 2int

Ω

V (119905) V119905119905(119905) d119909

+ 21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+ 2 (nabla119906 (119905) nabla119906

119905(119905))

+ 2 (nablaV (119905) nablaV119905(119905)) + 2120573

(50)


(119906119905119905(119905) 119906 (119905)) + (V

119905119905(119905) V (119905)) + (nabla119906 (119905) nabla119906

119905(119905))

+ (nablaV (119905) nablaV119905(119905))

= minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2minus119872(nablaV (119905)2

2) nablaV (119905)2

2

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

(51)

which implies

11987110158401015840(119905) = 2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus 2119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 2120573

minus 2int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus 2int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(52)

Therefore we have

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2+ 120573)


minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)


1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)


Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0


119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)


(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)



120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3gt


3) = 119864(0) Set 120572

0= (1198971nabla11990602

2+ 1198972nablaV02

2)12


3) which



0isin (0 119879)

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12

lt 1205723 (44)


2

2+ 1198972nablaV(119905)2


0

such that

(1198971

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV (1199050)1003817100381710038171003817

2

2)

12



119864 (1199050) ge 119866 [(119897

1

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nabla119906 (1199050)1003817100381710038171003817

2

2)

12

]

gt 119866 (1205723) = 119864 (0)

(46)



119871 (119905) 11987110158401015840(119905) minus (1 + 120590) 119871

1015840(119905)2ge 0 (47)


time 119879lowastle 119871(0)[120590119871



lowast119871(119905) = infin


defined by

119871 (119905) = 119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591

+ (119879 minus 119905) (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) + 120573(119905 + 119879

0)2

(48)



1198711015840(119905) = 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ nabla119906 (119905)2

2+ nablaV (119905)2

2minus (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

+ 2120573 (119905 + 1198790)

= 2int

Ω

119906 (119905) 119906119905(119905) d119909 + 2int

Ω

V (119905) V119905(119905) d119909

+ 2int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+ 2int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591 + 2120573 (119905 + 119879

0)

(49)

and consequently

11987110158401015840(119905) = 2int

Ω

119906 (119905) 119906119905119905(119905) d119909 + 2int

Ω

V (119905) V119905119905(119905) d119909

+ 21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+ 2 (nabla119906 (119905) nabla119906

119905(119905))

+ 2 (nablaV (119905) nablaV119905(119905)) + 2120573

(50)


(119906119905119905(119905) 119906 (119905)) + (V

119905119905(119905) V (119905)) + (nabla119906 (119905) nabla119906

119905(119905))

+ (nablaV (119905) nablaV119905(119905))

= minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2minus119872(nablaV (119905)2

2) nablaV (119905)2

2

minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

(51)

which implies

11987110158401015840(119905) = 2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus 2119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 2120573

minus 2int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

minus 2int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(52)

Therefore we have

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

minus119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2+ 120573)


minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)


1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)


Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0


119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)


(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)



120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909) minus 2 (119901 + 3)

times [int

Ω

119906 (119905) 119906119905(119905) d119909 + int

Ω

V (119905) V119905(119905) d119909 + 120573 (119905 + 119879

0)

+ int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

+int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591]

2

= 2119871 (119905) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2minus119872(nabla119906 (119905)

2

2) nabla119906 (119905)

2

2


2+ 120573)

minus 2119871 (119905) (int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+ int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

minus2 (119901 + 2)int

Ω

119865 (119906 V) d119909)

+ 2 (119901 + 3) 119867 (119905)


1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

timesΨ (119905)

(53)


Ψ (119905) =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

+ int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591 + 120573

119867 (119905) = (119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591 + 120573(119905 + 119879

0)2

)Ψ (119905)

minus [int

Ω

[119906 (119905) 119906119905(119905) + V (119905) V

119905(119905)] d119909

+ int

119905

0


119905(120591))] d120591

+120573 (119905 + 1198790) ]

2

(54)


(int

Ω

119906 (119905) 119906119905(119905) d119909)

2

le 119906 (119905)2

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


(int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591)

2

le int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

Ω

V (119905) V119905(119905) d119909

le 119906 (119905)2

1003817100381710038171003817V119905(119905)

10038171003817100381710038172

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172V (119905)

2

le

1

2

119906 (119905)2

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2V (119905)2

2

(55)

Similarly we have

int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591


int

Ω

119906 (119905) 119906119905(119905) d119909int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

119906 (119905)2

2int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2int

119905

0

nablaV (120591)22d120591

int

Ω

V (119905) V119905(119905) d119909int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le

1

2

V (119905)22int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2int

119905

0

nabla119906 (120591)2

2d120591

int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591int

119905

0

(nablaV (120591) nablaV119905(120591)) d120591

le

1

2

int

119905

0

nabla119906 (120591)2

2d120591int119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

+

1

2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591int119905

0

nablaV (120591)22d120591

(56)



120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120573 (119905 + 1198790) int

Ω

119906 (119905) 119906119905(119905) d119909

le 120573 (119905 + 1198790) 119906 (119905)

2

1003817100381710038171003817119906119905(119905)

10038171003817100381710038172

le

1

2

120573119906 (119905)2

2+

1

2

120573(119905 + 1198790)21003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2


120573 (119905 + 1198790) int

119905

0

(nabla119906 (120591) nabla119906119905(120591)) d120591

le 120573 (119905 + 1198790) int

119905

0

nabla119906 (120591)2

1003817100381710038171003817nabla119906119905(120591)

10038171003817100381710038172d120591

le

1

2

120573int

119905

0

nabla119906 (120591)2

2d120591 + 1

2

120573(119905 + 1198790)2

int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591



[0 119879] Using (53) we get

119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 119871 (119905)Φ (119905) (58)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2(int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

+int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909)

minus 2 (119901 + 2) 120573

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

(59)


minus int

Ω

int

119905

0

1198921(119905 minus 119904) Δ119906 (119904) 119906 (119905) d119904 d119909

= int

119905

0

1198921(119905 minus 119904) int

Ω


= int

119905

0

1198921(119905 minus 119904) int

Ω


+ (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(60)

Similarly

minus int

Ω

int

119905

0

1198922(119905 minus 119904) ΔV (119904) V (119905) d119904 d119909

= int

119905

0

1198922(119905 minus 119904) int

Ω


+ (int

119905

0

1198922(119904) ds) nablaV (119905)2

2

(61)


Φ (119905) = minus 2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

+ 2int

119905

0

1198921(119904) dsnabla119906 (119905)2

2+ 2int

119905

0

1198922(119904) dsnablaV (119905)2

2

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909 minus 2 (119901 + 2) 120573

+ 2int

119905

0

1198921(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591

+ 2int

119905

0

1198922(119905 minus 119904) int

Ω


minus 2 (119901 + 3)int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591

(62)

Since we have

2int

119905

0

119892119894(119905 minus 119904) int

Ω


ge minus120576int

119905

0


2

2d119904

minus

1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2


1

120576

(int

119905

0

119892119894(119904) d119904) nabla119906 (119905)

2

2 119894 = 1 2

(63)



Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Φ (119905) ge minus2 (119901 + 2) (1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2)

minus 120576 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ (2 minus

1

120576

)

times [int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2+ int

119905

0

1198922(119904) d119904nablaV (119905)2

2]

minus 2 (119901 + 3) (int

119905

0

1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2d120591 + int

119905

0

1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2d120591)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2


2

minus 2 (119901 + 2) 120573 minus 4 (119901 + 2) 119864 (119905) + 4 (119901 + 2) 119864 (119905)

= minus4 (119901 + 2) 119864 (119905) + 2 (119901 + 2)119872(nabla119906 (119905)2

2)

minus 2119872(nabla119906 (119905)2

2) nabla119906 (119905)

2

2

+ 2 (119901 + 2)119872(nablaV (119905)22)


2

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198921(119904) d119904nabla119906 (119905)2

2

minus 2 (119901 + 2) 120573

minus 2 (119901 + 1 +

1

2120576

)int

119905

0

1198922(119904) d119904nablaV (119905)2

2

minus 2 (119901 + 3)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(64)


Φ (119905) ge minus4 (119901 + 2) 119864 (0) + 2 (119901 + 1)


119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

minus

1

120576

(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ 2 (119901 + 1) (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

minus

1

120576

(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2minus 2 (119901 + 2) 120573

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

= minus4 (119901 + 2) 119864 (0)

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2minus 2 (119901 + 2) 120573

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

120576

)int

119905

0

1198922(119904) d119904]


+ [2 (119901 + 2) minus 120576] [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(65)


Φ (119905) ge [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2

+ [2 (119901 + 1)1198980

minus(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) (minus2119864 (0) minus 120573) ge 0

(66)



we get

Φ (119905)

ge minus4 (119901 + 2) 1205751198641

+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198921(119904) d119904]

times nabla119906 (119905)2

2


+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ [2 (119901 + 1)1198980minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)

times int

119905

0

1198922(119904) d119904]


+ 2 (119901 + 1)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 1) 120575 [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(67)


int

119905

0

119892119894(119904) d119904

le int

infin

0

119892119894(119904) d119904

le

2 (119901 + 1)1198980

2 (119901 + 1) + 1 2 [(1 minus 120575)2(119901 + 2) + 120575 (1 minus 120575)]

=

2 (119901 + 1)1198980minus 2120575 (119901 + 1)119898

0

2 (1 minus 120575) (119901 + 1) + 1 2 (1 minus 120575) (119901 + 2) + 2120575

(119894 = 1 2)

(68)

that is

2120575 (119901 + 1) [1198980minus int

119905

0

119892119894(119904) d119904]

le 2 (119901 + 1)1198980

minus (2 (119901 + 1) +

1

2 (119901 + 2) (1 minus 120575) + 2120575

)int

119905

0

119892119894(119904) d119904

(69)


Φ (119905) ge minus 4 (119901 + 2) 1205751198641

+ 2120575 (119901 + 1) (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

(70)

Since

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12



(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

12

ge 1205723 (72)


2 (119901 + 2)

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

ge

119901 + 1

2 (119901 + 2)

1205722

3gt

119901 + 1

2 (119901 + 2)

1205722

lowastgt 1198641

(73)


Φ (119905) ge 4 (119901 + 2) [1205751198641minus 1205751198641] ge 0 (74)


119871 (119905) 11987110158401015840(119905) minus

119901 + 3

2

1198711015840(119905)2ge 0 (75)



(119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

gt1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2gt 0

(76)

consequently

1198711015840(0) = 2int

Ω

11990601199061d119909 + 2int

Ω

V0V1d119909 + 2120573119879

0gt 0 (77)


119879 gt (10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1205731198792

0)

times ((119901 + 1) (int

Ω

11990601199061d119909 + int

Ω

V0V1d119909 + 120573119879

0)

minus (1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) )

minus1

gt 0

(78)



lowastle



lowast119871(119905) =


lim119905rarr119879

minus

lowast

(nabla119906 (119905)2

2+ nablaV (119905)2

2) = infin (79)






11986110158401015840(119905) minus 4 (120575

lowast+ 1) 119861

1015840(119905) + 4 (120575

lowast+ 1) 119861 (119905) ge 0 (80)

If

1198611015840(0) gt 119903

2119861 (0) + 119870

0 (81)

then

1198611015840(119905) gt 119870

0(82)


2= 2(120575

lowast+ 1) minus


1199032minus 4 (120575

lowast+ 1) 119903 + 4 (120575

lowast+ 1) = 0 (83)




lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra












lowast

for 119905 ge 0 (84)






119879lowastle 2(3120575lowast+1)2120575


radic119886lowast

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(85)


lowast


119870 (119905) = 119870 (119906 (119905) V (119905)) = 119906 (119905)2

2+ V (119905)2

2

+ int

119905

0

nabla119906 (120591)2

2d120591 + int

119905

0

nablaV (120591)22d120591 119905 ge 0

(86)


1198701015840(119905) gt

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 119905 gt 0 (87)


1198701015840(119905) = 2int

Ω

119906119906119905d119909 + 2int

Ω

VV119905d119909 + nabla119906

2

2+ nablaV2

2 (88)

11987010158401015840(119905) = 2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ 2

1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ 2int

Ω

119906119906119905119905d119909

+ 2int

Ω

VV119905119905d119909 + 2int

Ω


+ 2int

Ω


(89)


11987010158401015840(119905) = 2 (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus 2119872(nabla119906

2

2) nabla119906

2

2


2+ 4 (119901 + 2)int

Ω

119865 (119906 V) d119909

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(90)

By (27) (29) and (90) we get

11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0)

+ 4 (119901 + 2)int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2) [119872(nabla1199062

2) +119872(nablaV2

2)]

minus [2119872(nabla1199062

2) + 2 (119901 + 2)int

119905

0

1198921(119904) d119904] nabla1199062

2

minus [2119872(nablaV22) + 2 (119901 + 2)int

119905

0

1198922(119904) d119904] nablaV2

2

+ 2 (119901 + 2) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

minus 2 (119901 + 2)int

119905

0

[(1198921015840

1 nabla119906) (120591) + (119892

1015840

2 nablaV) (120591)] d120591

+ 2int

119905

0

int

Ω


+ 2int

119905

0

int

Ω


(91)


2int

119905

0

int

Ω


= 2int

119905

0

1198921(119905 minus 119904) int

Ω


+ 2 (int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

ge minus2 (119901 + 2) (1198921 nabla119906) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

(92)

Similarly

2int

119905

0

int

Ω


ge minus2 (119901 + 2) (1198922 nablaV) (119905)

+ (2 minus

1

2 (119901 + 2)

)(int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

(93)



11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

+ 2 (119901 + 2)119872(nabla1199062

2)

minus [2119872(nabla1199062

2)

+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198921(119904) d119904]

timesnabla1199062

2

+ 2 (119901 + 2)119872(nablaV22)


+(2 (119901 + 1) +

1

2 (119901 + 2)

)int

119905

0

1198922(119904) d119904]

times nablaV22

(94)


11987010158401015840(119905) minus 2 (119901 + 3) (

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

ge minus4 (119901 + 2) 119864 (0) + 4 (119901 + 2)

times int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(95)

We note that

nabla119906 (119905)2

2minus1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2= 2int

119905

0

int

Ω



1003817100381710038171003817

2

2= 2int

119905

0

int

Ω


(96)


nabla119906 (119905)2

2+ nablaV (119905)2

2le1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

+ int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(97)


1198701015840(119905) le 119870 (119905) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2

+ int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(98)


11987010158401015840(119905) minus 2 (119901 + 3)119870

1015840(119905) + 2 (119901 + 3)119870 (119905) + 119871

4ge 0 (99)

where

1198714= 4 (119901 + 2) 119864 (0) + 2 (119901 + 3) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (100)

Let

119861 (119905) = 119870 (119905) +

1198714

2 (119901 + 3)

119905 gt 0 (101)


1198701015840(0) gt (119901 + 3 minus radic(119901 + 3) (119901 + 1)) [119870 (0) +

1198714

2 (119901 + 3)

]

+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2

(102)

that is if

119864 (0) lt

radic119901 + 3


int

Ω

(11990601199061+ V0V1) d119909

minus

119901 + 3

2 (119901 + 2)

(10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

(103)


1015840(119905) gt nabla119906

02

2+nablaV02

2 119905 gt




ℎ (119905) = [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

minus120575lowast

for 119905 isin [0 1198791]

(104)



ℎ1015840(119905) = minus120575

lowastℎ(119905)1+1120575

lowast

(1198701015840(119905) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2) (105)

ℎ10158401015840(119905) = minus120575

lowastℎ(119905)1+2120575

lowast

119881 (119905) (106)

where

119881 (119905) = 11987010158401015840(119905) [119870 (119905) + (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

(107)



119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875 = 119906 (119905)2

2+ V (119905)2

2

119876 = int

119905

0

(nabla119906 (120591)2

2+ nablaV (120591)2

2) d120591

119877 =1003817100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2+1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

119878 = int

119905

0

(1003817100381710038171003817nabla119906119905(120591)

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905(120591)

1003817100381710038171003817

2

2) d120591

(108)


1198701015840(119905) le 2 (radic119875119877 + radic119876119878) +

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2 (109)

By (95) we get

11987010158401015840(119905) ge (minus4 minus 8120575

lowast) 119864 (0) + 4 (1 + 120575

lowast) (119877 + 119878) (110)



lowast) (119877 + 119878)]

times [119870 (119905) + (1198791minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]


2

(111)



minus1120575lowast

+ 4 (1 + 120575lowast) (119877 + 119878) (119879

1minus 119905) (

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)


2

]

(112)



minus1120575lowast

119905 ge 0 (113)


ℎ10158401015840(119905) le 120575

lowast(4 + 8120575

lowast) 119864 (0) ℎ(119905)

1+1120575lowast

119905 ge 0 (114)


ℎ1015840(119905)2ge 1198861+ 1198871ℎ(119905)2+1120575

lowast

for 119905 ge 0 (115)

where

1198861= 120575lowast2ℎ(0)2+2120575

lowast

times [(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2


]

(116)

1198871= 8120575lowast2119864 (0) (117)


119864 (0) lt

(1198701015840(0) minus

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2minus1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

2

8 [119870 (0) + 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

=

(int

Ω

11990601199061d119909 + int

Ω

V0V1d119909)2

2 [10038171003817100381710038171199060

1003817100381710038171003817

2

2+1003817100381710038171003817V0

1003817100381710038171003817

2

2+ 1198791(1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)]

(118)


lim119905rarr119879


119879lowastle 2(3120575lowast+1)2120575


radic1198861

1 minus [1 + 119888ℎ (0)]minus12120575

lowast

(119)

Therefore

lim119905rarr119879

lowastminus

119870 (119905)

= lim119905rarr119879

lowastminus

(119906 (119905)2

2+ V (119905)2

2+ int

119905

0

nabla119906 (120591)2

2d120591

+int

119905

0

nablaV (120591)22d120591) = infin

(120)



1198791ge 119879lowast




1and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (121)

Then

(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

12

lt 120572lowast

(122)

for all 119905 isin [0 119879)



119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868 (119905) = 119868 (119906 (119905) V (119905)) = (1198980minus int

119905

0

1198921(119904) d119904) nabla119906 (119905)

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

119869 (119905) = 119869 (119906 (119905) V (119905))

=

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+

1

2

[ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

+

1198981

120574 + 1

(nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)]

minus int

Ω

119865 (119906 V) d119909

(123)


1

and

(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (124)

Then

119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2) le

2 (119901 + 2)

119901 + 1

119864 (119905) (125)

(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905) (126)

for all 119905 isin [0 119879)


(1198971

1003817100381710038171003817nabla1199060

1003817100381710038171003817

2

2+ 1198972

1003817100381710038171003817nablaV0

1003817100381710038171003817

2

2)

12

lt 120572lowast (127)


1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

lt 1205722

lowast= 120578minus1(119901+1)

1

(128)

which implies that

119868 (119905) ge (1198980minus int

infin

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

infin

0

1198922(119904) d119904) nablaV2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

= 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus (119886119906 + V2(119901+2)2(119901+2)

+ 2119887119906V119901+2119901+2

)

ge 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

minus 1205781(1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)

119901+2

= (1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2)


2

2+ 1198972nablaV (119905)2

2)

119901+1

] ge 0

(129)


119869 (119905) ge

1

2

[(1198980minus int

119905

0

1198921(119904) d119904) nabla119906

2

2

+ (1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2+ (1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) ]

minus int

Ω

119865 (119906 V) d119909 minus

1

2 (119901 + 2)

119868 (119905) +

1

2 (119901 + 2)

119868 (119905)

ge (

1

2

minus

1

2 (119901 + 2)

)


119905

0

1198921(119904) d119904) nabla119906

2

2

+(1198980minus int

119905

0

1198922(119904) d119904) nablaV2

2]

+ (

1

2

minus

1

2 (119901 + 2)

) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2 (119901 + 2)

119868 (119905)


ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ge

119901 + 1

2 (119901 + 2)

times [1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2

+ (1198921 nabla119906) (119905) + (119892

2 nablaV) (119905) ]

+

1

2 (119901 + 2)

119868 (119905)

ge 0

(130)


119897 (nabla119906 (119905)2

2+ nablaV (119905)2

2)

le 1198971nabla119906 (119905)

2

2+ 1198972nablaV (119905)2

2le

2 (119901 + 2)

119901 + 1

119869 (119905)

le

2 (119901 + 2)

119901 + 1

119864 (119905)

(1198921 nabla119906) (119905)

+ (1198922 nablaV) (119905) le

2 (119901 + 2)

119901 + 1

119869 (119905) le

2 (119901 + 2)

119901 + 1

119864 (119905)

(131)


1198661(119905) = 119864 (119905) + 120576

1120601 (119905) + 120576

2120595 (119905) (132)


120601 (119905) = 1205851(119905) int

Ω

119906 (119905) 119906119905(119905) d119909 + 120585

2(119905) int

Ω

V (119905) V119905(119905) d119909

(133)

120595 (119905) = minus 1205851(119905) int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus 1205852(119905) int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

(134)


1 (119894 = 1 2) in (133) and (134)


2such

that

12057311198661(119905) le 119864 (119905) le 120573

21198661(119905) forall119905 ge 0 (135)


1003816100381610038161003816le 1199062

1003817100381710038171003817119906119905

10038171003817100381710038172

+ V2

1003817100381710038171003817V119905

10038171003817100381710038172

le

1

2

(1199062

2+1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+ V22+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le

1

2

1198622(nabla119906

2

2+ nablaV2

2) +

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

le 1198622 119901 + 2

119897 (119901 + 1)

119864 (119905) + 119864 (119905) = (1 + 1198622 119901 + 2

119897 (119901 + 1)

)119864 (119905)

(136)


int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

le1003817100381710038171003817119906119905

10038171003817100381710038172

times [int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909]12

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

int

Ω

(int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2

+

1

2

(1198980minus 119897) int

Ω

int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904))

2d119904 d119909

le

1

2

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

1 nabla119906) (119905)

(137)


int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

le

1

2

1003817100381710038171003817V119905(119905)

1003817100381710038171003817

2

2+

1

2

1198622(1198980minus 119897) (119892

2 nablaV) (119905)

(138)


1003816100381610038161003816120595 (119905)

1003816100381610038161003816le

1

2

(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+

1

2

1198622(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le 119864 (119905) +

119901 + 2

119901 + 1

1198622(1198980minus 119897) 119864 (119905)

= (1 +

119901 + 2

119901 + 1

1198622(1198980minus 119897))119864 (119905)

(139)


from (132) (136) and (139)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894(119905) equiv 1 (119894 = 1 2)) satisfies

1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)] nabla1199062

2


1

2

)] nablaV22

+

1

4120598

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d 119909


2(120574+1)

2+ nablaV2(120574+1)


(140)



1206011015840(119905) =

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2+ int

Ω

119906119906119905119905d119909 + int

Ω

VV119905119905d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


2) nabla119906

2


2) nablaV2

2

+ int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

+ int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

minus int

Ω


Ω


times int

Ω

119865 (119906 V) d119909

(141)


int

Ω


119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904 d119909

le (int

119905

0

1198921(119905 minus 119904) d119904) nabla119906 (119905)

2

2

+ int

Ω

int

119905

0

1198921(119905 minus 119904) |nabla119906 (119905)|


119905

0

1198921(119904) d119904)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) le (1 + 120598) (119898

0minus 119897)

times nabla119906 (119905)2

2+

1

4120598

(1198921 nabla119906) (119905) forall120598 gt 0

(142)


int

Ω


0

1198922(119905 minus 119904) nablaV (119904) d119904 d119909

le (1 + 120598) (1198980minus 119897) nablaV (119905)2

2+

1

4120598

(1198922 nablaV) (119905)

(143)

int

Ω


120598

2

nabla1199062

2+

1

2120598

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2 (144)

int

Ω


120598

2

nablaV22+

1

2120598

1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2 forall120598 gt 0 (145)


1206011015840(119905) le

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2minus 1198980nabla1199062

2


2(120574+1)

2+ nablaV2(120574+1)

2)

+ (1 + 120598) (1198980minus 119897) (nabla119906

2

2+ nablaV (119905)2

2)

+

1

4120598

((1198921 nabla119906) (119905) + (119892

2 nablaV) (119905))

+

120598

2

(nabla1199062

2+ nablaV2

2) +

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909

=1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2


1

2

)] nabla1199062

2minus [119897 minus 120598 (119898

0minus 119897 +

1

2

)]

times nablaV22+

1

4120598

(1198921 nabla119906) (119905) +

1

4120598

(1198922 nablaV) (119905)

+

1

2120598

(1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)


2(120574+1)

2+ nablaV2(120574+1)

2)

+ 2 (119901 + 2)int

Ω

119865 (119906 V) d119909 forall120598 gt 0

(146)



119894(119905) equiv 1 (119894 = 1 2)) satisfies

1205951015840(119905) le [120578 (119898

0minus 119897) (3119898

0minus 2119897) + 3119862120578119862

7]

times (nabla1199062

2+ nablaV2

2) + 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+ 1205781198981(1198980minus 119897) (nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ [(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus

1198920(0)

4120578

1198622[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+ (120578 minus int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905

1003817100381710038171003817

2

2

+ (120578 minus int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905

1003817100381710038171003817

2

2 forall120578 gt 0

(147)

where

1198920(0) = max 119892

1(0) 119892

2(0)

1198626= 1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

1198627= 1198622(2119901+3)

[

2 (119901 + 2)

119897 (119901 + 1)

119864 (0)]

2(p+1)

(148)



1205951015840(119905) = minusint

Ω

119906119905119905(119905) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905119905(119905) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909

minus int

Ω

119906119905(119905) int

119905

0

1198921(119905 minus 119904) 119906

119905(119905) d119904 d119909

minus int

Ω

V119905(119905) int

119905

0

1198922(119905 minus 119904) V

119905(119905) d119904 d119909

= int

Ω

119872(nabla1199062

2) nabla119906 (119905)

sdot int

119905

0


+ int

Ω

119872(nablaV22) nablaV (119905)

sdot int

119905

0


minus int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

times [int

119905

0


minus int

Ω

(int

119905

0

1198922(119905 minus 119904) nablaV (119904) d119904)

times [int

119905

0


+ [int

Ω

nabla119906119905(119905) int

119905

0


+int

Ω

nablaV119905(119905) int

119905

0


minus [int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

1198912(119906 V) int

119905

0

1198922(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus [int

Ω

119906119905(119905) int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904 d119909

+int

Ω

V119905(119905) int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904 d119909]

minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

= 1198681+ sdot sdot sdot + 119868

7minus (int

119905

0

1198921(119904) d119904) 1003817

100381710038171003817119906119905(119905)

1003817100381710038171003817

2

2

minus (int

119905

0

1198922(119904) d119904) 1003817

100381710038171003817V119905(119905)

1003817100381710038171003817

2

2

(149)




10038161003816100381610038161198681

1003816100381610038161003816le 120578119872(nabla119906

2

2) nabla119906

2

2int

119905

0

1198921(119904) d119904

+

1

4120578

119872(nabla1199062

2) (1198921 nabla119906) (119905)

= 120578 (1198980nabla1199062

2+ 1198981nabla1199062(120574+1)

2) int

119905

0

1198921(119904) d119904

+

1

4120578

(1198980+ 1198981nabla1199062120574

2) (1198921 nabla119906) (119905)

le 120578 (1198980minus 119897) (119898

0nabla1199062

2+ 1198981nabla1199062(120574+1)

2)

+

1

4120578

[1198980+ 1198981(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

120574

] (1198921 nabla119906) (119905)

le 1205781198980(1198980minus 119897) nabla119906

2

2+ 1205781198981(1198980minus 119897) nabla119906

2(120574+1)

2

+

1198626

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(150)


Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Similarly we have

10038161003816100381610038161198682

1003816100381610038161003816le 1205781198980(1198980minus 119897) nablaV2

2+ 1205781198981(1198980minus 119897) nablaV2(120574+1)

2

+

1198626

4120578

(1198922 nablaV) (119905) forall120578 gt 0

(151)



10038161003816100381610038161198683

1003816100381610038161003816le 120578int

Ω

(int

119905

0

1198921(119905 minus 119904) nabla119906 (119904) d119904)

2

d119909

+

1

4120578

int

Ω

(int

119905

0


2

d119909

le 120578int

Ω

[int

119905

0


2

d119909

+

1

4120578

(1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(int

119905

0

1198921(119904) d119904)

2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905)

le 2120578(1198980minus 119897)2

nabla1199062

2

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

1 nabla119906) (119905) forall120578 gt 0

(152)

Similarly

10038161003816100381610038161198684

1003816100381610038161003816le 2120578(119898

0minus 119897)2

nablaV22

+ (2120578 +

1

4120578

) (1198980minus 119897) (119892

2 nablaV) (119905) forall120578 gt 0

(153)


10038161003816100381610038161198685

1003816100381610038161003816le 120578 (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1

4120578

(1198980minus 119897) [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(154)


10038161003816100381610038161003816100381610038161003816

int

Ω

1198911(119906 V) int

119905

0

1198921(119905 minus 119904) (119906 (119905) minus 119906 (119904)) ds dx

10038161003816100381610038161003816100381610038161003816

le 120578int

Ω

(119886|119906 + V|2119901+3 + 119887|119906|119901+1

|V|119901+2)2

d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578int

Ω

(|119906|2(2119901+3)

+ |V|2(2119901+3) + |119906|2(119901+1)

|V|2(119901+2)) d119909

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 119862120578 [1199062(2119901+3)

2(2119901+3)+ V2(2119901+3)2(2119901+3)

+ 1199062(119901+1)

2(2119901+3)V2(119901+2)2(2119901+3)

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

times [nabla1199062(2119901+3)

2+ nablaV2(2119901+3)

2+ nabla119906

2(119901+1)

2nablaV2(119901+2)2

]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le 1198621205781198622(2119901+3)

(

2 (119901 + 2)

119897 (119901 + 1)

119864 (0))

2119901+2

times [nabla1199062

2+ nablaV2

2+ nabla119906

2nablaV2]

+

1198622(1198980minus 119897)

4120578

(1198921 nabla119906) (119905)

le

3

2

1198621205781198627[nabla119906

2

2+ nablaV2

2]

+

C2 (1198980minus 119897)

4120578

(1198921 nabla119906) (119905) forall120578 gt 0

(155)

Hence we infer that

10038161003816100381610038161198686

1003816100381610038161003816le 3119862120578119862

7(nabla119906

2

2+ nablaV2

2)

+

1198622(1198980minus 119897)

4120578

[(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall120578 gt 0

(156)


10038161003816100381610038161198687

1003816100381610038161003816le 120578

1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

1(119905 minus 119904) (119906 (119905) minus 119906 (119904)) d119904)

2

d119909

+ 1205781003817100381710038171003817V119905

1003817100381710038171003817

2

2+

1

4120578

int

Ω

(int

119905

0

1198921015840

2(119905 minus 119904) (V (119905) minus V (119904)) d119904)

2

d119909

le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) +

1198622

4120578

(int

119905

0

1198921015840

1(119904) d119904) (119892

1015840

1 nabla119906) (119905)

+

1198622

4120578

(int

119905

0

1198921015840

2(119904) d119904) (119892

1015840

2 nablaV) (119905)


le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 120578 (1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2) minus

1198921(0) 1198622

4120578

(1198921015840

1 nabla119906) (119905)

minus

1198922(0) 1198622

4120578

(1198921015840

2 nablaV) (119905) forall120578 gt 0

(157)



0gt

0

int

119905

0

119892119894(119904) d119904 ge int

1199050

0

119892119894(119904) d119904 (119894 = 1 2) 119905 ge 119905

0 (158)

denote

1198923= minint

1199050

0

1198921(119904) d119904 int

1199050

0

1198922(119904) d119904 (159)


0 we have

1198661015840

1(119905) = 119864

1015840(119905) + 120576

11206011015840(119905) + 120576

21205951015840(119905)

le minus1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7]

times [nabla1199062

2+ nablaV2

2]


times (nabla1199062(120574+1)

2+ nablaV2(120574+1)

2)

minus (1 minus

1205761

2120598

minus 1205762120578) (

1003817100381710038171003817nabla119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817nablaV119905

1003817100381710038171003817

2

2)

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]


1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)

+ 21205761(119901 + 2) int

Ω

119865 (119906 V) d119909

+ (

1

2

minus

1198920(0)

4120578

12057621198622)

times [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

(160)


1205762120578 (1198980minus 119897) lt 120576

1lt 1205762(1198923minus 120578) (161)


1198628= 12057621198923minus 1205762120578 minus 1205761gt 0

11986210

= 11989811205761minus 11989811205762120578 (1198980minus 119897) gt 0

11986211

=

1

2

minus

1198920(0)

4120578

12057621198622gt 0

11986212

= 1 minus

1205761

2120598

minus 1205762120578 gt 0

(162)

We choose 120578 1205761 and 120576


further

1198629= 1205761[119897 minus 120598 (119898

0minus 119897 +

1

2

)]

minus 1205762120578 [(1198980minus 119897) (3119898

0minus 2119897) + 3119862119862

7] gt 0

(163)

So we arrive at

1198661015840

1(119905) le minus 119862

8(1003817100381710038171003817119906119905

1003817100381710038171003817

2

2+1003817100381710038171003817V119905

1003817100381710038171003817

2

2)


2

2+ nablaV2

2) + 2120576

1(119901 + 2)int

Ω

119865 (119906 V) d119909

minus 11986210(nabla119906

2(120574+1)

2+ nablaV2(120574+1)

2)

+ 11986211[(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]

+

1205761

4120598

+ 1205762[(2120578 +

2 + 1198622

4120578

) (1198980minus 119897) +

1198626

4120578

]

times [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

(164)


1198661015840

1(119905) le minus119888

lowast119864 (119905) + 119862 [(119892

1 nabla119906) (119905) + (119892

2 nablaV) (119905)]

forall119905 ge 1199050

(165)


and (28) that

120585 (119905) 1198661015840

1(119905) le minus119888

lowast120585 (119905) 119864 (119905)

+ 119862120585 (119905) [(1198921 nabla119906) (119905) + (119892

2 nablaV) (119905)]

le minus119888lowast120585 (119905) 119864 (119905) + 119862120585

1(119905) (1198921 nabla119906) (119905)

+ 1198621205852(119905) (1198922 nablaV) (119905)

le minus119888lowast120585 (119905) 119864 (119905)

minus 119862 [(1198921015840

1 nabla119906) (119905) + (119892

1015840

2 nablaV) (119905)]


1015840(119905) forall119905 ge 119905

0

(166)

That is

1198711015840


lowast120585 (119905) 119864 (119905) le minus119896120585 (119905) 119871

lowast(119905) forall119905 ge 119905

0 (167)


where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 119871lowast(119905) = 120585(119905)119866



119871lowast(119905) le 119871

lowast(1199050) 119890minus119896int119905

1199050120585(119904)d119904

forall119905 ge 1199050 (168)




Acknowledgments


References














































119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























119905119905=



119905119905= minus119860119906 + F(119906)rdquo









Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article general decay and blow-up of solutions for ...decay rate of the solution energy is...

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