on the blow-up of the solution of an equation related to the hamilton-jacobi equation

ISSN 0001-4346, Mathematical Notes, 2013, Vol. 93, No. 1, pp. 90–101. © Pleiades Publishing, Ltd., 2013.Original Russian Text © M. O. Korpusov, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 1, pp. 81–95.

On the Blow-Up of the Solution of an EquationRelated to the Hamilton–Jacobi Equation

M. O. Korpusov*

Moscow State UniversityReceived March 26, 2012; in final form, June 8, 2012

Abstract—A new model three-dimensional third-order equation of Hamilton–Jacobi type is de-rived. For this equation, the initial boundary-value problem in a bounded domain with smoothboundary is studied and local solvability in the strong generalized sense is proved; in addition,sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time)solvability are obtained.

DOI: 10.1134/S0001434613010100

Keywords: third-order equation of Hamilton–Jacobi type, blow-up of solutions, electric po-tential in a crystalline semiconductor, Dirichlet problem, Galerkin approximation, Browder–Minty theorem, Lipschitz-continuous operator.

1. INTRODUCTION

Consider the initial boundary-value problem for the following model equation:

∂

∂t(Δu − u) + Δu + μ|∇u|p = 0, p > 0, μ > 0. (1.1)

Note that the Hamilton–Jacobi equation has the following form:

∂u

∂t− μ|∇u|2 = 0. (1.2)

Therefore, it is possible to call (1.1) an equation related to the Hamilton–Jacobi equation. Theclassical equation (1.2) was studied by many authors and we shall not present any relevant results forit. Let us now consider some other equations of Hamilton–Jacobi type. First, consider the Kuramoto–Sivashinsky equation

∂u

∂t+ Δu + Δ2u +

12|∇u|2 = 0, (1.3)

which was studied by many authors. Also note recent results obtained in [1]–[4]. These papers dealtwith the initial and initial boundary-value problems for the Kuramoto–Sivashinsky equation in boundedand unbounded domains and present important results on global (in time) solvability and blow-up infinite time. In these papers, the method of nonlinear capacity developed in the papers of Pokhozhaev andMitidieri [5] was used. In addition, the theory of thermal explosion [6] involves an equation of the form

∂T

∂t= ΔT + |∇T |2, (1.4)

where T = T (x, t) is the temperature. Note, however, (see [6]) that, by the replacement

v = exp(T ),

*E-mail: [email protected]

90

BLOW-UP OF THE SOLUTION OF A HAMILTON–JACOBI EQUATION 91

Eq. (1.4) can be reduced to the linear heat equation

∂v

∂t= Δv.

Numerous papers are devoted to the study of nonlinear parabolic equations of the form

∂u

∂t= Δu + μ|∇u|p + f(x, u), (1.5)

for which results concerning blow-up and global (in time) solvability [6]–[10] were obtained.

2. DERIVATION OF EQ. (1.1)

Consider a domain Ω ⊂ RN for N ≥ 3 with smooth boundary ∂Ω ∈ C

2,δ, δ ∈ (0, 1], whose surface issimply connected. Suppose that this domain is filled with a crystalline semiconductor. It is well knownthat, in the theory of semiconductors (see, for example, [11], [12]), quasistationary approximation is oftensuccessfully used for the system of Maxwell equations [13], whose electric part has the form

div D = −4πn, rotE = 0, D = E + 4πP, (2.1)

where D is the electric displacement vector, E is the electric field-strength vector, P is the polarizationvector of the medium, and n is the free charge density; further, in view of the fact that the surface of theboundary ∂Ω is simply connected, the electric field potential φ is expressed as

E = −∇φ.

Since the semiconductor is, obviously, a conducting medium, we must supplement Eqs. (2.1) with theequation for the free charge current, which has the following form:

∂n

∂t= divJ, (2.2)

where J is the free charge current vector. Naturally, we need to write an expression for this vector. Takinginto account thermal heating of the semiconductor [11], we can write [13]:

J = σE − γ∇T, σ > 0, γ > 0, (2.3)

where T is the semiconductor temperature. Further, in view of the thermal heating due to the electricfield E, the temperature T satisfies the heat equation [11]:

ε∂T

∂t= ΔT + Q(|E|), (2.4)

where the parameter ε > 0 has the following form [11]:

ε = ε0e−α; (2.5)

here ε0 > 0 is a fixed number and the parameter α > 0 is sufficiently large. The function Q(|E|) describesthermal pumping in the semiconductor in a self-consistent electric field E and is well approximated bythe power function [11]:

Q(|E|) = q0|E|p, p > 0, q0 > 0. (2.6)

By the “smallness” property of the parameter (2.5), instead of Eq. (2.4) we consider the followingequation:

ΔT + Q(|E|) = 0. (2.7)

Now note that, by its physical meaning, the quantity

divP = ρ

describes the distribution of bound charges in the semiconductor and is well approximated by the linearfunction

ρ = ρ0φ, ρ0 > 0. (2.8)

MATHEMATICAL NOTES Vol. 93 No. 1 2013

92 KORPUSOV

The differential consequence of system of equations (2.1)–(2.8) is the following equation:

∂

∂t(Δφ − 4πρ0φ) + 4πσΔφ − 4πγq0|∇φ|p = 0, (2.9)

which, by elementary replacements, can be reduced to the required equation of Sobolev type

∂

∂t(Δφ − φ) + Δφ + μ|∇φ|p = 0, p > 0, μ > 0. (2.10)

In the following sections, we consider either the Dirichlet problem, or the Neumann problem;therefore, let us discuss the physical meaning of these boundary conditions. Indeed, the homogeneousDirichlet condition

φ|∂Ω = 0

means that the boundary of the semiconductor is “grounded” (by convention, the electric potential ofthe earth is zero). The homogeneous Neumann condition

∂φ

∂nx

∣∣∣∣∂Ω

= 0

means that the normal component of the electric field at the boundary of the semiconductor is zero.

3. THE DIRICHLET PROBLEM IN Ω ⊂ RN FOR N ≥ 3

First, let us present the classical statement of the problem in its generalized setting. Thus, letΩ ⊂ R

N for N ≥ 3 be a bounded domain with smooth boundary ∂Ω ∈ C2,δ for δ ∈ (0, 1]. Consider the

following problem:

∂

∂t(Δu − u) + Δu + μ|∇u|p = 0 in QT , (3.1)

u = 0 in ΓT , u(x, 0) = u0(x) in Ω, (3.2)

where QT = (0, T ) × Ω and ΓT = (0, T ) × ∂Ω, and consider the case in which p > 0.Now let us compare problem (3.1), (3.2) with its generalized setting.

Definition 1. A function u(x)(t) of class

u(x)(t) ∈ L∞(0,T; H1

0(Ω)), u′(x)(t) ∈ L2(0,T; H1

0(Ω)) (3.3)

is called the weak generalized solution of problem (3.1)–(3.2) if, for some T > 0, the followingequality holds:

〈D(u), w〉 = 0 for a.e. t ∈ [0, T ], for all w ∈ H10(Ω), (3.4)

where 〈 · , · 〉 is the the duality bracket between the Hilbert spaces H10(Ω) and H

−1(Ω), while

D(u) ≡ ∂

∂t(Δu − u) + Δu + μ|∇u|p.

and u0(x) ∈ H10(Ω).

The case p ∈ (0, 1]. First, we obtain the first a priori estimate, which immediately implies thatthere is no blow-up in this case. But if we require greater smoothness of the initial functionu0(x) ∈ H

10(Ω) ∩ H

2(Ω), then we shall prove both global (in time) solvability in the sense of Definition 1and even greater smoothness.

Thus, first, in relation (3.4), we take the function w = u(x)(t); then, after integrating by parts, weobtain the following equality:

12

d

dt[‖∇u‖2

2 + ‖u‖22] + ‖∇u‖2

2 = μ

ˆΩ|∇u|pu dx, (3.5)



where we use the notation

‖v‖q ≡(ˆ

Ω|v|q dx

)1/q

for q ≥ 1.

Since, for p ∈ (0, 1], it is obvious that p ≤ 1 + 2/N for N ≥ 3, it follows that, in view of Holder’sinequality and the embedding

H10(Ω) ⊂ L

q(Ω) for q =2

2 − p, (3.6)

the following chain of estimates holds:∣∣∣∣μ

ˆΩ|∇u|pu dx

∣∣∣∣≤ |μ|

(ˆΩ|∇u|2 dx

)p/2(ˆΩ|u|2/(2−p)

)(2−p)/2

≤ c1(Ω, p)|μ|(ˆ

Ω|∇u|2 dx

)(p+1)/2

,

where c1 = c1(Ω, p) is the corresponding constant of the embedding (3.6). Combining this with (3.5),we obtain the inequality

12

d

dt[‖∇u‖2

2 + ‖u‖22] ≤ c1|μ|‖∇u‖p+1

2 . (3.7)

If the notation

Φ(t) ≡ 12[‖∇u‖2

2 + ‖u‖22],

is used, then inequality (3.7) easily yields the inequality

dΦdt

≤ c2Φα, c2 = c1|μ|2(p+1)/2, α =p + 1

2. (3.8)

Consider separately the following two cases: p = 1 and p ∈ (0, 1). Indeed, in the first case, inequal-ity (3.8) implies the estimate

Φ(t) ≤ Φ0ec2t, Φ0 ≡ Φ(0). (3.9)

In the second case, using (3.8), we obtain the inequality

Φ(t) ≤ (Φ(1−p)/20 + c2t)2/(1−p), Φ0 ≡ Φ(0). (3.10)

Thus, it follows from inequalities (3.9) and (3.10) that the weak generalized solution of prob-lem (3.1), (3.2) does not blow up in finite time for p ∈ (0, 1]. Let us prove that the weak generalizedsolution exists globally in time for u0(x) ∈ H

10(Ω)∩H

2(Ω). To do this, we shall use the classical Galerkinmethod in combination with the compactness method [14].

Step 1. Statement of the Galerkin approximation problem. In the Hilbert space H10(Ω), take an

(orthonormal in L2(Ω)) basis {wj} ⊂ H

10(Ω) composed of the solutions of the eigenfunction problem

Δwj + λjwj = 0, wj ∈ H10(Ω),

and consider the Galerkin approximations

um =m∑

j=1

cmj(t)wj , (3.11)

for which the following problem is stated:

〈D(um), wj〉 = 0 j = 1, . . . ,m, (3.12)

where

D(um) ≡ ∂

∂t(Δum − um) + Δum + μ|∇um|p.


94 KORPUSOV

In addition, consider the initial conditions

cmj(0) = αj , um(0) =m∑

j=1

αjwj → u0 strongly in H10(Ω) as m → +∞. (3.13)

Step 2. Local solvability of the Galerkin approximations. In view of the special choice of thebasis {wj}, the system of Galerkin approximations (3.12) can easily be reduced to the following form:

(1 + λj)dcmj

dt+ λjcmj = Fj(cm), Fj(cm) = μ

ˆΩ

∣∣∣∣

m∑

k=1

cmk∇wk

∣∣∣∣

p

wj , (3.14)

where cm = (cm1, . . . , cmm) and j = 1, . . . ,m. By Peano’s theorem, the system of ordinary differentialequations (3.14) with the initial condition (3.13), has at least one solution of class

cmj(t) ∈ C(1)[0, Tm] for some Tm > 0, j = 1, . . . ,m. (3.15)

Step 3. A priori estimates. We already have the first a priori estimates (3.9) and (3.10), valid for theGalerkin approximations um, from which we see that

‖∇um‖22(t) ≤ c(T ) < +∞ for all t ∈ [0, T ], (3.16)

where T > 0 is an arbitrary fixed number. Note that, in view of the special choice of the basis {wj}, thefollowing chain of equalities holds:

‖∇um‖22 =

ˆΩ|∇um|2 dx =

m,m∑

k,j=1,1

cmjcmk

ˆΩ(∇wj ,∇wk) dx = −

m,m∑

k,j=1,1

cmjcmk

ˆΩ

wjΔwk dx

=m,m∑

k,j=1,1

cmjcmkλk

ˆΩ

wjwk dx =m∑

k=1

λkc2mk(t).

It follows from this chain of equalities and the first a priori estimate (3.16) that the time Tm > 0 ofexistence of the solution of system (3.14) is independent of m ∈ N.

Let us derive the second a priori estimate. To do this, we multiply both sides of relation (3.12) by c′mj

and sum over j = 1, . . . ,m. Then, after integrating by parts, we obtain the following equality:

‖∇u′m‖2

2 + ‖u′m‖2

2 +12

d

dt‖∇um‖2

2 = μ

ˆΩ|∇um|pu′

m dx, (3.17)

The following estimates hold:∣∣∣∣μ

ˆΩ|∇um|pu′

m dx

∣∣∣∣≤ |μ|

(ˆΩ|∇um|2 dx

)p/2(ˆΩ|u′

m|2/(2−p) dx

)(2−p)/2

≤ |μ|c1(Ω, p)‖∇um‖p2‖∇u′

m‖2 ≤ ε

2‖∇u′

m‖22 +

μ2c21

2ε‖∇um‖2p

2 . (3.18)

For ε = 1, relations (3.17) and (3.18) imply the second a priori estimateˆ T

0‖∇u′

m‖22 dt ≤ c(T ) < +∞ (3.19)

for all t ∈ [0, T ] and for an arbitrary fixed T > 0. Unfortunately, in order to pass to the limit inrelation (3.12), we need the third and fourth a priori estimates. To this end, note that since ∂Ω ∈ C

2,δ forδ ∈ (0, 1], it follows that wj ∈ H

10(Ω) ∩ H

2(Ω). Now suppose that

u0(x) ∈ H10(Ω) ∩ H

2(Ω), um(0) → u0 strongly in H10(Ω) ∩ H

2(Ω). (3.20)

On the other hand, since

wj = − 1λj

Δwj ,



relation (3.12) implies the equality

〈D(um),Δwj〉 = 0, j = 1, . . . ,m, (3.21)

where

D(um) ≡ ∂

∂t(Δum − um) + Δum + μ|∇um|p.

Let us now multiply both sides of relation (3.21) by cmj and sum over j = 1, . . . ,m; then, afterintegrating by parts, we obtain the equality

12

d

dt[‖Δum‖2

2 + ‖∇um‖22] + ‖Δum‖2

2 = μ

ˆΩ|∇um|pΔum dx. (3.22)

The following inequality holds:∣∣∣∣μ

ˆΩ|∇um|pΔum dx

∣∣∣∣≤ |μ|

(ˆΩ|Δum|2 dx

)1/2(ˆΩ|∇um|2p dx

)1/2

. (3.23)

Consider separately the following two cases: p = 1 and p ∈ (0, 1). In the first case, we easily obtain theinequality

∣∣∣∣μ

ˆΩ|∇um|pΔum dx

∣∣∣∣≤ 1

2‖Δum‖2

2 +|μ|22

‖∇um‖22 . (3.24)

In the second case, the following inequality holds:(ˆ

Ω|∇um|2p dx

)1/2

≤ |Ω|(1−p)/2‖∇um‖p2; (3.25)

combining this with (3.23), we obtain the following inequality:∣∣∣∣μ

ˆΩ|∇um|pΔum dx

∣∣∣∣≤ 1

2‖Δum‖2

2 + c3‖∇um‖2p2 , where c3 =

|Ω|1−pμ2

2, (3.26)

and |Ω| is the Lebesgue measure on the domain Ω. n the case under consideration p ∈ (0, 1], usinginequalities (3.22), (3.24), and (3.26), we obtain the third a priori estimate:

‖Δum‖22 + ‖∇um‖2

2 ≤ c(T ) < +∞ (3.27)

for all t ∈ [0,T] for an arbitrary fixed T > 0.Finally, let us obtain the fourth a priori estimate. To do this, let us multiply both sides of relation (3.21)

by c′mj and sum over j = 1, . . . ,m. Then, after integrating by parts, we obtain the following equality:

‖Δu′m‖2

2 + ‖∇u′m‖2

2 +12

d

dt‖Δum‖2

2 = μ

ˆΩ|∇um|pΔu′

m dx. (3.28)

Now arguing just as in the derivation of the second a priori estimate (3.19), we obtain the fourth a prioriestimate ˆ T

0[‖Δu′

m‖22 + ‖∇u′

m‖22] dt ≤ c(T ) < +∞ (3.29)

for all t ∈ [0,T] and for any fixed T > 0.

Step 4. Passage to the limit. In view of well-known results, it follows from the a priori esti-mates (3.27) and (3.29) that there exists a subsequence {umm} ⊂ {um} and a function u(x)(t) suchthat the following properties are valid:

umm

∗⇀ u ∗-weakly in L

∞(0, T ; H10(Ω) ∩ H

2(Ω)), (3.30)

u′mm

⇀ u′ weakly in L2(0, T ; H1

0(Ω) ∩ H2(Ω)), (3.31)

for any fixed T > 0.Note that the following important statement actually proved in [14] (see also [15]) holds.


96 KORPUSOV

Lemma 1. Let the sequence {um}+∞m=1 satisfy the following conditions uniformly in m ∈ N:

‖um‖V0 ≤ c,

ˆ T

0dt ‖u′

m ‖2V0≤ c for all t ∈ (0, T ), (3.32)

where 0 < T < +∞ and c > 0 is a constant depending on T , but independent of m ∈ N. Inaddition, let the completely continuous embedding

V0 ↪→ W

be valid. Then there exists a function u(s) ∈ W for almost all s ∈ [0, T ] and the following limitequality holds:

umk(s) → u(s) strongly in W for almost all s ∈ [0, T ].

Take V0 = H10(Ω) ∩ H

2(Ω), in the assertion/statement of this theorem; for W = H10(Ω) we find that

umm(s) → u(s) strongly in H10(Ω) for almost all s ∈ [0, T ]. (3.33)

Now we can pass to the limit as m → +∞ in relation (3.12). The passage to the limit in the expressionˆΩ|∇umm |pwj dx (3.34)

needs a special study. Note that, for p ∈ (0, 1], the following inequality holds:

|ap − bp| ≤ |a − b|p, a > 0, b > 0.

Therefore, the following inequality is valid:∣∣∣∣

ˆΩ|∇umm |pwj dx −

ˆΩ|∇u|pwj dx

∣∣∣∣≤ˆ

Ω

∣∣|∇umm |p − |∇u|p

∣∣|wj| dx ≤

ˆΩ|∇umm −∇u|p|wj | dx

≤(ˆ

Ω|∇umm −∇u|2 dx

)p/2(ˆΩ|wj |2/(2−p) dx

)(p−2)/p

; (3.35)

hence, in view of (3.33), we obtain the required result for each fixed j ∈ N. Thus, the following statementis valid.

Theorem 1. Let p ∈ (0, 1], and let u0(x) ∈ H10(Ω) ∩ H

2(Ω). Then there exists a weak generalizedsolution of problem (3.1), (3.2) of class

u(x)(t) ∈ L∞(0, T ; H1

0(Ω) ∩ H2(Ω)), u′(x)(t) ∈ L

2(0, T ; H10(Ω) ∩ H

2(Ω))

for an arbitrary T > 0.

The case 1 ≤ p ≤ 1 + 2/N . First, let us prove the theorem on the local solvability of prob-lem (3.1)–(3.2) regarded in the weak generalized sense of Definition 1 for the class

u(x)(t) ∈ C(1)([0,T0); H1

0(Ω))

for some maximal time T0 > 0 depending on u0(x) ∈ H10(Ω). To this end, consider the following

operator:

Au ≡ −Δu + I: H10(Ω) → H

−1(Ω). (3.36)

Using the Browder–Minty theorem in the same way as in [15], we can prove that this operator isinvertible and the inverse operator is Lipschitz-continuous with Lipschitz constant equal to 1. Then,for the class

u(x)(t) ∈ C(1)([0,T]; H1

0(Ω)),

the initial boundary-value problem (3.1), (3.2) regarded in the weak sense of Definition 1 can beexpressed in the following abstract form:

dv

dt+ B1A

−1v = μ|B2A−1v|p, v(0) = v0 = A

−1u0, v = A−1u, (3.37)



where

B1 ≡ −Δ: H10(Ω) → H

−1(Ω), B2 ≡ ∇ : H10(Ω) → L

2(Ω) ⊗ · · · ⊗ L2(Ω). (3.38)

Note that the operator

B1A−1v : H

−1(Ω) → H−1(Ω) (3.39)

is Lipschitz-continuous. Indeed, the following chain of inequalities holds:

‖B1A−1v1 − B1A

−1v2‖∗ ≤ ‖A−1v1 − A−1v2‖ ≤ ‖v1 − v2‖∗ for all v1, v2 ∈ H

−1(Ω). (3.40)

Let us now prove the bounded Lipschitz-continuity of the operator |B2A−1v|p. Indeed, for q = 2/p

and p ∈ [1, 2), the following chain of inequalities holds:∥∥|B2A

−1v1|p − |B2A−1v2|p

∥∥∗

≤ p∥∥|∇A

−1(v1) −∇A−1(v2)|max{|∇A

−1(v1)|p−1, |∇A−1(v2)|p−1}

∥∥∗

≤ c1(p)(ˆ

Ω|∇A

−1(v1) −∇A−1(v2)|q

× max{

|∇A−1(v1)|q(p−1), |∇A

−1(v2)|q(p−1)}

dx

)1/q

≤ c1(p)(ˆ

Ω|∇A

−1(v1) −∇A−1(v2)|2 dx

)1/2

×(ˆ

Ωmax

{

|∇A−1(v1)|2(p−1)/(2−q), |∇A

−1(v2)|2(p−1)/(2−q)}

dx

)(2−q)/(2q)

= μ(R)‖∇A−1(v1) −∇A

−1(v2)‖2, (3.41)

where

μ(R) = c1Rp−1, R = max{

‖∇A−1(v1)‖2, ‖∇A

−1(v2)‖2

}

.

Now note that, for the class

v(t) ∈ C(1)([0,T]; H−1(Ω)),

we can reduce our problem (3.37) to the integral equation

v(t) = v0 +ˆ t

0ds G(v)(s), G(v)(s) = −B1A

−1v(s) + μ|B2A−1v(s)|p, (3.42)

and, for a sufficiently small T > 0, in view of estimates (3.40) and (3.41), we can apply the contractionmapping method in the Banach space

L∞(0,T; H−1(Ω))

and prove the existence of a unique solution of the integral equation (3.42) in this Banach space. Further,we can use the boot-strap method (see, for example, [15]) and prove that this solution is, in fact, fromthe class

C(1)([0,T]; H−1(Ω)).

Further, we must apply the standard algorithm for the continuation of solutions of integral equationswith respect to time, obtaining the following result.

Theorem 2. Let p ∈ [1, 2) and u0(x) ∈ H10(Ω). Then there exists a T0 = T0(u0) > 0 such that there

is a unique solution u(x)(t) of problem (3.1), (3.2) regarded in the weak sense of Definition 1 forthe class

u(x)(t) ∈ C(1)([0,T0); H1

0(Ω));

here T0 is either T0 = +∞ or T0 < +∞ and, in the latter case, the following limit equality holds:

lim supt↑T0

‖∇u‖2(t) = +∞. (3.43)


98 KORPUSOV

Now let us take up the derivation of the a priori estimate from which it will follow that, for sufficientlysmall initial data, we have T0 = +∞. Indeed, in relation (3.4), we take for w ∈ H

10(Ω) the solution

u(x)(t) ∈ C(1)([0,T0); H1

0(Ω))

itself and then, after integrating by parts, we obtain the following equality:

dΦdt

+ ‖∇u‖22 = μ

ˆΩ|∇u|pu dx, Φ(t) ≡ 1

2‖∇u‖2

2 +12‖u‖2

2. (3.44)

By Friedrichs’ inequality,

λ1‖u‖22 ≤ ‖∇u‖2

2

it follows from (3.44) the inequality

dΦdt

+ λ1‖u‖22 ≤ μ

ˆΩ|∇u|pu dx. (3.45)

Using (3.44) and (3.45), we obtain the inequality

dΦdt

+ 2λ1

λ1 + 1Φ(t) ≤ μ

ˆΩ|∇u|pu dx. (3.46)

Let us now estimate the integral on the right-hand side of the last inequality. Indeed, let

p ∈[

1,1 + 2N

]

for N ≥ 3.

Then the following chain of inequalities holds:ˆ

Ω|∇u|pu dx ≤

(ˆΩ|∇u|2 dx

)p/2(ˆΩ|u|2/(2−p) dx

)(p−2)/2

≤ c2(Ω)(ˆ

Ω|∇u|2 dx

)p/2(ˆΩ|∇u|2 dx

)1/2

= c2(Ω)‖∇u‖p+12 ≤ c2(Ω)2(p+1)/2Φ(p+1)/2(t). (3.47)

Thus, it follows from (3.46) and (3.47) that

dΦdt

+ 2λ1

λ1 + 1Φ(t) ≤ c3Φ(p+1)/2, c3 = c22(p+1)/2|μ|. (3.48)

This inequality is easily integrated, and we obtain the upper bound

Φ(t) ≤ e−αt

[Φ(1−p)/20 − (c3/α)[1 − e−α(p−1)t/2]]2/(p−1)

, α = 2λ1

λ1 + 1, c3 = c22(p+1)/2|μ|. (3.49)

Under the condition

Φ0 ≤(

α

c22(p+1)/2|μ|

)2/(p−1)

, Φ0 = Φ(0), α = 2λ1

λ1 + 1, (3.50)

inequality (3.49) implies the estimate

Φ(t) ≤(

α

c22(p+1)/2|μ|

)2/(p−1)

. (3.51)

Thus, we have proved the following statement.



Theorem 3. For N ≥ 3, let p ∈ [1, 1 + 2/N ], let u0(x) ∈ H10(Ω), and let the following inequality

hold:

Φ0 ≤(

α

c22(p+1)/2|μ|

)2/(p−1)

, Φ0 =12‖∇u0‖2

2 +12‖u0‖2

2, α = 2λ1

λ1 + 1.

Then there exists a unique solution u(x)(t) of problem (3.1), (3.2) regarded in the weak sense ofDefinition 1 for the class

u(x)(t) ∈ C(1)([0,+∞); H1

0(Ω)).

The case p > p0. Before passing to the detailed study of this case, we need to study the convergenceof the following integral (the nonlinear “capacity”):

a = a(p; Ω;N) ≡ 1λ1

(ˆΩ

|∇ψ1|p/(p−1)

ψ1/(p−1)1 (x)

dx

)(p−1)/p

, (3.52)

where the function ψ1(x) is the first eigenfunction corresponding to the first eigenvalue λ1 > 0 of theLaplace operator in a bounded domain Ω ⊂ R

N with smooth boundary ∂Ω ∈ C2,δ for δ ∈ (0, 1]:

Δψ1(x) + λ1ψ1(x) = 0, ψ1(x)|∂Ω = 0. (3.53)

Let p0 = p0(Ω;N) > 1 denote a number such that, for p > p0, the integral on the right-hand side ofrelation (3.52) converges. Let us prove that there exist domains Ω ⊂ R

N , for which such a p0 exists.Indeed, let Ω = BR = {x ∈ R

N : |x| < R} be a ball of radius R > 0. Then we can calculate the firsteigenfunction and the first eigenvalue of problem (3.53) (see, for example, [6]). Indeed,

ψ1(x) = ψ1(|x|) = c0r(2−N)/2J(N−2)/2(λ

1/21 r), λ1 =

(zN1)2

R2, r = |x|,

where zN1 is the first root of the Bessel function J(N−2)/2(x). Now note that the Bessel function Jν(z)satisfies Euler’s formula [16]

Jν(z) =((1/2)z)ν

Γ(ν + 1)

+∞∏

n=1

{

1 − z2

z2ν,n

}

, (3.54)

where zν,1 < zν,2 < · · · < zν,n < · · · are the roots of the Bessel function Jν(z). It follows from theexplicit form (3.54) that the integrand in (3.52) has an integrable singularity for p > 2, i.e., the numberp0 = 2 in the case of the ball. Now suppose that we consider a domain Ω for which the number p0 exists,and let

p > p0. (3.55)

We obtain sufficient conditions for the blow-up of the weak generalized solution of problem (3.1), (3.2)of class

u(x)(t) ∈ C(1)([0, T ]; H1

0(Ω))

for such numbers p. To this end, in relation (3.4), for w we take ψ1(x) ∈ H10(Ω). Then, for this class,

after integrating by parts, we obtain the equality

(λ1 + 1)dJdt

+ λ1J(t) = μ

ˆΩ|∇u|pψ1(x) dx, J(t) ≡

ˆΩ

u(x)(t)ψ1(x). (3.56)

The following chain of expressions is valid:

|J| =∣∣∣∣

ˆΩ

uψ1(x) dx

∣∣∣∣=

1λ1

∣∣∣∣

ˆΩ

uΔψ1 dx

∣∣∣∣=

1λ1

∣∣∣∣

ˆΩ(∇u,∇ψ1) dx

∣∣∣∣

≤ 1λ1

ˆΩ|∇u||∇ψ1| dx =

1λ1

ˆΩ

ψ1/p1 |∇u| |∇ψ1|

ψ1/p1

dx


100 KORPUSOV

≤ 1λ1

(ˆΩ

|∇ψ1|p/(p−1)

ψ1/(p−1)1 (x)

dx

)(p−1)/p(ˆΩ|∇u|pψ1 dx

)1/p

= a

(ˆΩ|∇u|pψ1 dx

)1/p

. (3.57)

Thus, using inequalities (3.56) and (3.57), we obtain the following ordinary differential inequality:

(λ1 + 1)dJdt

+ λ1J(t) ≥ μ

ap|J(t)|p, J(t) ≡

ˆΩ

u(x)(t)ψ1(x), (3.58)

which can easily be reduced to the inequality

dΨdt

≥ βe−α(p−1)t|Ψ(t)|p, Ψ(t) = eαtJ(t), α =λ1

λ1 + 1, β =

μ

ap(λ1 + 1), (3.59)

which, in turn, implies the following inequality:

J(t) ≥ e−αt

[J1−p0 − (β/α)[1 − e−α(p−1)t]]1/(p−1)

, J0 = J(0). (3.60)

Now let the following initial condition hold:

J0 >

(λ1a

p

μ

)1/(p−1)

. (3.61)

Then inequality (3.60) implies the existence of a T∞ such that

lim supt↑T∞

J(t) = +∞, T∞ ≤ − 1p − 1

ln(

1 − λ1ap

μJ1−p

0

)

. (3.62)

Thus, we have proved the following statement.

Theorem 4. Let p > p0. Then, under the conditions

J0 >

(λ1a

p

μ

)1/(p−1)

, J0 ≡ˆ

Ωu0(x)ψ1(x) dx,

the following limit equality holds:

lim supt↑T∞

J(t) = +∞, T∞ ≤ − 1p − 1

ln(

1 − λ1ap

μJ1−p

0

)

,

where

J(t) ≡ˆ

Ωu(x)(t)ψ1(x) dx.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research (grant no. 08-01-00376-a)and the Presidential Program for the Support of Young Doctors of Science (grant no. NSh–99.2009.1).

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on the blow-up of the solution of an equation related to the hamilton-jacobi equation

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