The abstract Cauchy problem for the fractional evolutionequationCitation for published version (APA):Bazhlekova, E. G. (1998). The abstract Cauchy problem for the fractional evolution equation. (RANA : reports onapplied and numerical analysis; Vol. 9814). Technische Universiteit Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGYDepartment of Mathematics and Computing Science

RANA 98-14July 1998

The abstract Cauchy problem for the fractional evolutionequation

by

E. Bazhlekova

Reports on Applied and Numerical AnalysisDepartment of Mathematics and Computing ScienceEindhoven University of TechnologyP.O. Box 5135600 MB EindhovenThe NetherlandsISSN: 0926-4507

.'

1

THE ABSTRACT CAUCHY PROBLEM FOR THE FRACTIONAL

EVOLUTION EQUATION

E. Bazhlekova*

Abstract

We give necessary and sufficient conditions for an unbounded closed operatorA in a Banach space X such that the abstract Cauchy problem for the fractionalevolution equation

DC>u = Au, 0 < 0:' < 1,

can be solved. We also present conditions on A such that the fractional timeevolution is holomorphic. A relation between the solutions of the problem fordifferent 0:', 0 < 0:' :::; 1, is obtained. In particular this relation shows that theproblem has a holomorphic solution whenever A generates just a Co-semigroup.Some examples are given.

Mathematics Subject Classification: 12H22, 26A33, 35F10, 47D06.Key Words and Phrases: fractional calculus, semigroup theory, Mittag-Leffler

function, Wright function.

1. Introduetion

Consider a linear closed operator A in a Banach space X with dense domainD(A) C X. Let 0 < 0:' < 1. Given x E X we investigate the following Cauchyproblem in X:

Dfu(t) = Au(t),

u(O) = x.

Here Dr is the regularized fractional derivative of order 0:', 0 < 0:' < 1,

Dfu(t) = Jl-c> :t u(t),

where Jf is the fractional integration operator defined by

1 tJfu(t) = f(0:') Jo

(t - r)c>-lu(r) dr, 0:' > O.

(1.1)

(1.2)

(1.3)

(1.4)

The connection between Dr and the Riemann-Liouville fractional derivative

(1.5)

2

is given by

Dfu(t) = Dnu(t) - u(O)). (1.6)

The definition (1.5) is the most commonly adopted in mathematical orientedpapers and text-books (see e.g. [16]). The definition (1.3)1 is more suitable forproblems of physical interest (see [9], [13]) where the conventional initial conditionsare expressed in terms of integer derivatives.

For a = 1 the problem (1.1-2) is the classical abstract Cauchy problem andthere is a vast amount of literature devoted to it and its equivalent formulation- the semigroup theory (see [6], [17], [22] and the references there). The casea = 2 has been investigated in detail in [7], higher order equations have been alsodiscussed. The abstract Cauchy problem for non-integer values of a is investigatedin [10] and [11] (for 0 < a < 1), [12] (for a > 0), [19] and [20] (for 0 < a :::; 2).More general evolutionary equations are studied in the recent monograph [18].

For integer a the following results are classical:The Cauchy problem (1.1-2) can be solved if and only if• a = 1: A is the generator of a strongly continuous (Co) semigroup (the

theorem of Hille- Yosida, see [6], [17], [22]);• a = 2: A is the generator of a strongly continuous cosine function (this

implies: A-generator of a holomorphic semigroup, see [6], p.l00);• a = 3,4, ...: A is a bounded operator (see [6], p.99).Roughly speaking, the set of "permitted" operators A "shrinks" as a increases.

We will show that this holds in some sense also for 0 < a :::; 1.Using ideas related to the theory of first and second order abstract differential

equations ([6], [7]) we give in Theorem 3.1. necessary and sufficient conditionsfor solvability of the Cauchy problem (1.1-2), extending the conditions of theHille-Yosida theorem from a = 1 to a E (0,1). (Let us note that Theorem3.1. also follows as a particular case of a more general theorem given in [18], p.43.) Theorem 4.1. gives a sufficient condition when the solution is analytic insome sector of the complex plane containing the semiaxis t > 0 and describesthe properties of the solution in this case. For a = 1 this result coincides with awell-known result concerning analytic semigroups (see [6], [17], [22]). Using therepresentations of the Mittag- Leffler functions ([4], p.5-6), we give in Theorem4.2. a relation between the solutions of (1.1-2) for different a, 0 < a :::; 1, interms of an auxiliary function. In particular this relation proves that if A isthe generator of a Co-semigroup then the problem (1.1-2) with 0 < a < 1 has ananalytic solution. Some examples show that the analytic properties of the solutioncan be achieved for operators A that do not generate Co-semigroup.

1 This (alternative to Riemann-Liouville) definition of fractional derivative was originallyintroduced by Caputo [1,2] in the late sixties and adopted by Caputo and Mainardi [3] in

the framework of the theory of Linear Viscoelasticity.

3

2. Preliminaries

We summarize some properties of the fractional integrals and derivatives (formore details see [9], [13] and [16]). Similarly to the ordinary differentiation andintegration we have:

DfJfu(t) = u(t)j JfDfu(t) = u(t) - u(O), 0 < a < 1. (2.1)

Simple results valid for t > 0 are:

JQt{3 = fCB + 1) tQ+{3 DQt{3 = r(;'1 + 1) t{3-Q a> 0 ;'1 > -1. (2.2)t r(;'1 + a + 1) , t r(;'1 - a +1)' ,

If ;'1-a+1 E {O, -1, -2, ...} then r(;'1+1)/r(;'1-a+1) = 0 in (2.2). In particularDft

Q- 1 = 0, Dfl = C

Q/f(1 - a), while Dfl = 0, a> O.

Using the properties of the Laplace transform it is not difficult to prove that

(2.3)

The Mittag-Leffier functions, defined by the following series

00 nOOn

EQ(z) = ~ r(a: + 1)' EQ,{3(z) = ~ f(a~ + 11)' a,;'1 > 0, Z E C (2.4)

are entire functions which provide a simple generalization of the exponential func­tion eZ = E1(z) and play an important role in the theory offractional differentialequations. Similarly to the differential equation d/dt(ewt ) = wewt the Mittag­Leffler function EQ(z) satisfies the more general differential relation

(2.5)

The most interesting properties of the Mittag-Leffler functions are associated withtheir Laplace integral

(2.6)

and with their asymptotic expansions as z --+ 00 (see [4],[5]). If 0 < a < 2, ;'1 > 0

1 1EQ,{3(z) = -;z(l-{3)/Q exp(zl/Q) +cQ,{3(z), Iarg zl s 2mr, (2.7)

1EQ,{3(z) = cQ,{3(z), Iarg( -z)/ < (1 - 2a)7l", (2.8)

where

(3.1)

4

3. The abstract Cauchy problem. Necessary and sufficient conditionsfor solvability.

DEFINITION 3.1. A strong solution of the abstract differential equation (1.1)in t ~ 0 (t > 0) is an X-valued function u(t) such that u(t) is continuous for t ~ 0,u(t) E D(A) and Dfu(t) is continuous for t ~ 0 (t > 0) and (1.1) is satisfied fort ~ 0 (t > 0).

DEFINITION 3.2. ([12]) A Cauchy problem is called solvable in a sharply uni­formly well-posed manner (SUWP-solvable) if:

(a) There exists a dense subspace D of X such that for any xED there existsa solution u(t) of (1.1) in t ~ 0 satisfying (1.2);

(b) There exist constants M ~ 1, w ~ 0, not dependending on u(t) and x,such that

Ilu(t)11 ~ Mewtllu(O) II, t ~ O.

for any solution u(t) of (1.1) in t ~ O.

DEFINITION 3.3. Let °< a ~ 1, M ~ 1, w ~ 0. An operator A is said tobelong to Ca(M,w) if the Cauchy problem (1.1-2) satisfies (a) and (b).

Denote Ca(w) = U{Ca(M,w); M ~ I}, Ca = U{Ca(w); W ~ O}. Let us notethat A E C1 iff A generates a Co-semigroup.

In analogy with the case a = 1 we define a propagator Pt of a SUWP­solvable Cauchy problem, Le. a bounded operator Pt such that u(t) = Ptx forany solution u(t) of (1.1-2).

Let e(A) denote the resolvent set and R()") = ()..I _A)-l the resolvent operatorof A, as usual.

THEOREM 3.1. Let 0 < a ~ 1, M ~ 1, w ~ O. Then A E Ca(M, w) iff R()..C<)exists in the half plane Re).. > wand

,II ()..a-l R()..a))(n)11 < M n. , Re).. > w, n = 0,1,... (3.2)

- (Re).. - w)n+l

Proof. Assume that the problem (1.1-2) is SUWP-solvable and let P? bethe corresponding propagator, so that

(3.3)

Fix).. E C with Re).. > wand define an operator on X by

S()")x = 1000

e-At Ptx dt, x EX.

Since the norm of the integrand is bounded by Mllxlle(w-ReA)t, S()") is a boundedoperator in X. Assume now that xED. Then Ptx is a solution of (1.1-2)

5

and this together with the closedness of A and (2.3) implies that S(A)X E D(A)and AS(A)X = X~S(A)X - Xl<-lx. Using again the closedness of A it followsS(A)X ~ D(A) and

(3.4)

(3.5)

In particular (3.4) shows that (Aa J - A) : D(A) ~ X is onto. It is injectiveas well. To see this assume there exists x E D(A) with Ax = AaX. Then from(2.5) u(t) = Ea(Aata)X is a solution of (1.1-2). Since Iarg(Aata)1 ~ !m!' whenReA> W ~ 0, t> 0, from (2.7) we have

1IEa(Aata)1 ~ _eRe.\t - E > Mewt , t > T,

a

that contradicts (3.1) unless x = 0. Thus we have proved that R(Aa ) exists whenReA> wand S(A) = Aa- l R(Aa), that is

Aa - l R(Aa)X = 100

e-.\tPtx dt, x E X.

After easily justified differentiation under the integral sign we obtain

(Aa- l R(Aa))(n)x = (-1t100

tne-.\tPtx dt, n = 0,1, ... , x E X, ReA> w,

that together with (3.3) gives (3.2).Suppose now that R(Aa) exists in the half plane ReA> wand the conditions

(3.2) hold. We begin by constructing certain smooth solutions of (1.1-2). Letx E D(Am) for some m > 2/11' and define u(t) as the inverse Laplace transform ofAa- l R(Aa)X. Using the well-known formula

R(A)X = ~x + :2 Ax + ... + A~ Am-lx + A~ R(A)Amx,

we have

ta t2a t(m-l)au(t) = x+ f(a + 1) Ax+ r(2a + 1) A

2x+ ...+ r((m _ 1)11' + 1) Amx+J(t), (3.6)

where

(3.7)

(3.8)

converges absolutely and uniformly for j3 > 1 with respect to t on compact subsetsof t ~ 0. Therefore, taking j3 = mil' and j3 = mil' - 1, we see that J(t) and J'(t)

6

are continuous for t ~ 0, from (1.3) the same is true for D? I(t), hence for u(t)and D?u(t).

Let now t = 0 and AR be the part of the circle 1).1 = R to the right of the lineRe). = w'. Then

C l w'+ioo Id).1 c. 1 Id).1 . 1

11/(0)11::; -2 -1'1 = - 11m -1'1 ::; C hm R -1 = 0,7r w'-ioo A rna 27r R-+oo AR A rna R-+oo rna

so that u(O) = x. It remains to check (1.1). Denote

1 l w'+ir eAt aE ().ata)Ir(t) = _. - a ).a-1R().a)Arnx d)', t > O.

27rZ w'-ir ).(3

Since the integrand is a holomorphic function with respect to ). for Re). > w wecan shift the path of integration to AR, R = Jr 2 +w12 • But from (2.7) it followsthat aEa().ata) = eAt + O((I).lt)-a) for Re). > w, t> 0, ). --+ 00. Hence

Taking f3 = ma we have

(3.10)

and after differintegration under the integral sign justified on the basis of theasymptotic expansion (2.7) for d/dtEa().ata) = ).ata- 1Ea,a().ata), the uniformconvergence of (3.8) for f3 = ma - 1 on compacts of t ~ 0 and Fubini's theoremwe obtain

Again from (3.9) with f3 = (m - 1)a we have

(3.11)

On the other hand A can be applied under the integral sign in (3.7) with theconvergence of the resulting integral, so that u(t) E D(A) and using (1.6), (2.2)and (3.11) we have

Dfu(t) - Au(t) =

7

t(m-l)c> 1 lw1+ioo At- Amx +- _e_ Ac>-l R(AC>)(AC>1- A)Amx dA = 0,

r((m - l)a + 1) 2rri w'-ioo Amc>

hence u(t) is a strong solution of (1.1-2) in t ~ 0 for x E D(Am).

To prove the inequality (3.1) we use the Post-Widder inversion formula:

LEMMA 3.1. Let u(t) be a X valued continuous function defined in t ~ 0such that u(t) = O(exp(-yt)) as t -+ 00 for some 'Y and let (£U)(A) be the Laplacetransform of u(t). Then

u(t) = lim (_1)n (!:)n+l (£u)(n) (!:)n--+oo n! t t

(3.12)

uniformly on compacts oft> O.Routine calculations show that u(t) satisfies the conditions of Lemma 3.1.

with 'Y = w' and (£U)(A) = Ac>-l R(AC»X. Then, from (3.12) and the inequalities(3.2) it follows

(n/t)n+l ( wt) -(n+l)IIu(t)11 ~ Mllxll lim (/ ) +1 =Mllxll lim 1 - - = Mllxllewt

.n--+oo n t - w n n--+oo n

Next we define Ptx = u(t) for x E D(Am), t ~ 0, where u(t) is the solution~btained. Since D(A) is dense in X and (?(A) i= 0 then D(Am) is dense in X andPt can be extended to a bounded operator in X without increase of norm, thatis (3.3) holds for Ptc>. The proof will be complete if we show that any solutionu(t) of (1.1-2) admits the representation u(t) = Ptc>u(O). Since (1.1) must hold fort = 0 there is no solution for x ~ D(A) and it is enough to establish that PtC>x isthe unique solution of (1.1-2) for any x E D(A).

Actually for the uniqueness of the solution a weaker condition suffices:

THEOREM 3.2. ([5]) If ReA exists on the half line A > 8 (8) 0) and

limsup(l/A) In II R(AC» II = 0..\--+00

(3.13)

then the solution of the problem (1.1-2) is unique.It is not difficult to check that the inequalities (3.2) imply (3.13) and so we

have uniqueness of the solution of (1.1-2).Next we show that Ptx satisfies (1.1-2) for any x E D(A). If A E (?(A) it

follows immediately from the definition of Pt that

(3.14)

By the usual continuity argument (3.14) must hold for all x E X. Noting thatD(A) = R(A)X, (3.14) implies PtD(A) ~ D(A), t ~ 0 and

(3.15)

8

If we apply Jf to both sides of (1.1) and use (1.2), (2.1) and (3.15) we have

P?x - x = J'tPtCX Ax, x E D(Am).

Applying now R()') to both sides of this equality and using (3.14) it follows

R()')(P?x - x) = J't PtCX AR()')x, x E D(Am).

Since on both sides bounded operators are applied to x, the last equality musthold for all x EX. Making use of the fact that R()') is one-to-one, we concludethat

P?x - x = J'tAPtCXx, x E D(A).

Therefore Ptcx X is the (unique) solution of (1.1-2).0

REMARK 3.1. Inequalities (3.2) follow from their real counterparts

The proof uses T~lor series expansion for the function ).cx-IR().CX).

COROLLARY 3.1. If A E C1 (w) then A E CCX(w 1/ cx ) for any 11' E (0,1).

Proof. By induction in n we obtain the following representation

n+l().CX-l R().cx))(n) = (-It L c'k,n).kcx-n-l R().cx)k, n = 0,1, ... ,

k=1

(3.16)

where c'k,n ~ 0 are constants, satisfying cl,o = 1, cI,n = (n - 11') cI,n-l' c'k,n =(n - kl1')c'k,n_l + l1'(k - l)c'k_l,n_l' k = 2, ... , n, C~+I,n = I1'nC~,n_l' n = 1,2, ....Then inequalities (3.2) follow using (3.16). Since Corollary 3.1. is a particularcase of the more general Theorem 4.2., we omit the details of the proof.O

From (3.5),(3.12) and (3.16) we obtain the following representation:

COROLLARY 3.2.

1 n+lP?x = lim, L c'k n (I - (tln)CX A)-k x,

n-+oo n. k=1 '(3.17)

where 0 < 11' ~ 1, ck,n are the constants in (3.16).Next we express the operator A in terms of the propagator of the Cauchy

problem (1.1-2).

PROPOSITION 3.1. If Pt is the propagator of (1.1-2) and x E D(A) then

PCXx - xAx = r (11' + 1) lim t .

t.j.O tcx(3.18)

(3.19)

9

Proof. For a function v(t) continuous in t ~ 0 we have: limt.j.o v(t) =f(a + 1) limt.j.o CO!Jfv(t). Taking v(t) = Dr ptO!x and using (1.1) and (2.1) weobtain (3.18).1:J

Incidentally (3.18) shows that ptO!x is not differentiable at t = 0+.The following result is an immediate consequence of Theorem 3.1.

THEOREM 3.3. Let A be an operator such that R(AO!) exists in the half-planeReA> w, Pt an operator-valued function strongly continuous in t ~ 0 and suchthat IIP?II ::; Mewt

, t ~ O. Assume that for each x EX

100

e-)..t Ptx dt = AO!-l R(AO!)X, ReA> w.

Then, A E CO!(w) and ptO! is the propagator of (1.1-2).

Proof. We obtain the inequalities (3.2) in the same way as from (3.5).Applying Theorem 3.1. it results that A E CO!. Let Pt be the propagator of(1.1-2). Then (3.19) holds for both PtO! and PtO! and Pt = PtO! follows from theuniqueness of the Laplace transform.D

4. Holomorphic solutions

THEOREM 4.1. Let 0 < a ::; 1. Assume Q(A) J I;s,O!a(7r/2 +8); A=J. O} for some 0 < 8::; 7r/2 and

IIR(A)II::; G/IAI, A E I;s,O!.

{A largAI <

(4.1)

(4.2)

Then, A E CO!(O) and the corresponding propagator P? has tile following addi­tional properties:

(a) PtO! can be extended analytically into the sector .6.s = {I argtl < 8; t =J. O};(b) IlPtll is uniformly bounded in every closed subsector ~S-e of .6.s;(c) PtO!x --t x as t -!- 0, t E ~S-e, x EX;(d) For any x EX, Ptx E D(A) and II APt II ::; M(l +CO!), t> 0;(e) For any x EX, PtO!x is a strong solution of (1.1-2) in t> O.

Proof. Let t E ~S-e for some E E (0,8) and fJ E (8 - E/2, 8), Q> 0. Set

Pt = 21

. r e)..tAO!-lR(AO!)dA,7rZ Jr

where r = {re- i(1I" /2+0); Q::; r < oo} U {Qei'P; - (7r /2 + 0) ::; <p ::; 7r/2 + fJ} U

{re i (11" /2+0); Q ::; r < oo} is oriented counterclockwise. If A E r then AO! E I;s,O!,and so R(AO!) exists and from (4.1)

(4.3)

10

Let (! = 1/1tl and a = sinc/2. Then from (4.2) and (4.3) it follows

IIPt'11 :s c reRe(At) Id>'1 :s c {':>O e-ar dr + C r eCos'{J d<p:S M. (4.4)21r Jf' 1>'1 1r J1 r 1r Jo

This estimate shows that the integral in (4.2) is absolutely convergent for t E .6.s-e ,

hence (a) and (b) are satisfied for Pt.Now fix >., Re>' > 0, and take e in (4.3) such that f2 < Re>'. Then, using (4.2)'

we have

100 ~ 1 1 100

e-AtPt' dt = -. j1Ci-1 R(j1Ci) e-(A-J1.)t dt dj1 =o 21rZ f' 0

-1. r j1Ci-1 R(j1Ci)X dj1 = >.Ci-1 R(>.Ci) ,21rz Jf' j1 - >.

where Fubini's theorem and Cauchy's integral formula in the region of the rightof r are used. So we have proved that the conditions of Theorem 3.3. are fulfilledwith w = O. Therefore A E CCi(O) and the corresponding propagator Pt = Pt.

Next for x E D(A) by Cauchy's theorem and Cauchy's integral formula ift E ~S-e, t.} 0

~ reAt>.-1R(>,Ci)Axd>'-+~ r>.-1R(>,Ci)Axd>'=0,21rzJf' 21rzJf'

that together with (b) implies (c).Now, for t > 0 and (! = lit, applying A to both sides of (4.2) and using the

identity AR(>') = >.R(>.) - I we have

APtCi = ~ reAt >.Ci-1 AR(>.Ci) d>" = ~ reAt >.2Ci-1 R(>.Ci) d>' _ CCi .

21r~ Jf' 21rZ Jf' f( 1 - Qi)

Taking (! = lit and estimating APt in the same way as in (4.4) we obtain (d).At the end, applying Jf to both sides of (1.1) we have

(4.5)

But from (d) JfPtA is a bounded operator for t > O. Then (4.5) holds for anyx E X, t > O. This implies (e).D

PROPOSITION 4.1. If (!(A) :J {>. : Re>' > O} and for some constant M

IIR(>']III :s MIRe>., Re>' > O. (4.6)

then for any Qi E (0,1) A satisfies the hypotheses of Theorem 4.1. with 0 < <5 <min{(1/Qi - 1)1r /2,1r/2}.

11

Proof. Fix 0' and 8 satisfying the conditions mentioned above. Then0'(11"/2+8) < 11"/2 and so Eo,s C e(A). Taking (3, 0'(11"/2+8) < (3 < 11"/2 we obtain

IIR(.\) II :::; R~.\ = IAI~S{;1 < IAI:S(3' .\ E Eo,s,

where {;1 = arg.\ and (4.1) holds with C = M/ cos (3.0The next theorem describes the relation between the solutions of the Cauchy

problem (1.1-2) for different values of 0', 0 < 0' :::; 1.

THEOREM 4.2. Let 0 < 0' < (3 :::; 1, 'Y = 0'/(3, w 2 o. If A E C13 (w) thenA E CO(w1h) and the following representation holds

where

Ptx = ~ [00 <1>-y (~) pfx ds, t> 0,fY Jo fY

(4.7)

(4.8)

is an entire function. Moreover Pt can be analytically extended to the sector~(I, {;1 = min{(l/'Y - 1)11"/2,11"/2} and for any x E X PtOx is a strong solution of(1.1-2) in t > O.

Proof. We summarize some properties of <P-y(z) (for more details see [4], p.5, [13], [15] and [21]). It is a particular case of a special function known as Wrightfunction ([5]), entire in the complex plane:

W(z; 'Y, 8) = f: If( zn +8) = 21

. [ fL-s exp(fL +ZfL--Y) dfL, 'Y > -1, 8 E C,n=O n. 'Yn 11"Z Jr

where r is a contour which starts at -00, encircles the origin once counter­clockwise and turns to its starting point. Obviously <1>-y(z) = W( -Z; -'Y, 1 - 'Y)and the integral representation

holds. For real values of Z we have ([4], p. 5):

<P-y(t) 2 0, t 2 o.

(4.9)

(4.10)

Let us denote IPt,-y(s) = (l/t-Y)<P-y(s/t-Y). Then the following formula is true ([4],p. 5):

(4.11)

12

From the asymptotic expansion of the Wright function given in [21] (this functionis called there generalized Bessel function) we obtain the inequality:

IW(-z;-')',8)1 ~ Bexp(-bzl/(l-.'t)) (4.12)

for Iargzl < min{(l - ')')3rr /2, rr}, Izl --t 00, 0 < ')' < 1, 8 E C, where B =B(')',8), b= b(')', 8) are positive constants.

Now define

(4.13)

Our aim is to prove that P? satisfies the conditions of Theorem 3.3. Since A E Cf3 ,there exist constants M ~ 1 and w ~ 0 such that

(4.14)

and together with (4.10) and (4.11) we obtain

(4.15)

We observe that a constant C'Y exists, such that

(4.16)

Indeed, from (2.7) there exists a constant C I such that E'Y(wt'Y) ~ C I exp(wlht)for every t ~ T. From the continuity of the Mittag-Leffler function in t ~ 0 itfollows that there exists a constant C 2 such that E'Y(wt'Y) ~ C 2 ~ C 2 exp(wlht)for t E [0, T]. Hence, we can take C'Y = max{C I , C2 }. Now from (4.15) and (4.16)we have

IlptOl1 ~ M' exp(wlht), t ~ O.

Taking,X E C with Re'x > wlh we shall prove that R('x°) exists. If,X = 1,Xleie

then 181 < rr/2 and for such 8 and')' E (0,1) the inequality cos ')'8 ~ cos'Y 0 holds.Then Re('x'Y) = I,XI'Y cos ')'0 ~ I,XI'Y cos'Y (} = (Re'xp > w. Since A E Cf3, it followsfrom Theorem 3.1. that R(pP) exists in the half-plane Rep, > w. Taking p, = ,X'Ywe obtain that R('x°) = R(p,f3) exists.

It is plain that P? is strongly continuous for t > O. To prove the strongcontinuity at the origin we substitute 0' = s/t'Y in (4.13). Then, using the identity1000

<I>'Y(0') dO' = 1 (that follows from (4.11) setting w = 0) and the strong continuity

of pf at the origin, we have

~ 100 100

lim Pt x = lim <I>'Y (0' )ptrr x dO' = <I>'Y (O')x dO' = x.t.j..O t.j..O 0 0

(4.17)

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For Re). > w1h, using (4.13) and interchanging the order of integration wehave 100

e->.tPt dt = 100

pf100

e->.tipt,'Y(S) dt ds. (4.18)

Substituting f.l = tr in (4.9) and shifting the new contour r' = r /t to r we getthe integral representation

Then, noting that Re(). - r) > 0 and using Cauchy's formula it follows

roo e->.tipt''Y(s)dt=~ r r'Y-1exp(-r'Ys) (>0 e-(>'-7)tdtdr=io 21l"Z ir io

-11 r'Y-1exp(-r'Ys) d _ \1'-1 (\"Y)- r - 1\ exp -1\ s .21l"i r r - ).

Using now (3.5) for pI, we obtain from (4.18) and (4.20)

(4.19)

(4.20)

100

e->.t PtQ dt = ).1'-1100

exp( _).1's)pf ds = ).1'-1 ).1'(13- 1 ) R().'YI3) = ).Q-1R().Q).

So we have proved the conditions of Theorem 3.3. Therefore A E CQ(w1h) andthe corresponding propagator Pt = Pt

Q•

Using the integral representation (4.19) we have

allIs-ipt (s) = - r'Y exp(rt - r"Y s) dr = -W(--' -, -,).at ,"I 21l"i r p+1 t'Y ' ,

Now, applying the inequalities (4.12) and (4.14) and noting that 1/{, - 1) > 1for 0 < , < 1 it easily follows that the integral in the right-hand side of (4.13) isabsolutely convergent for t E b.o and the differentiation under the integral sign ispossible, that is Pt admits analytic extention into b.o.

Differintegrating under the integral sign in (4.19) we obtain

Dfipt 'Y(s) = 21

. rrQ

+'Y-1exp(rt - r"Y s) dr = ~+W(-~; -" 1- a - ,)., 1l"Z i r p Q t'Y

Then, differintegrating (4.13) and applying again (4.12) and (4.14), it results thatDfPt is a bounded operator for t > O. Since A E CQ

that from (1.6),(2.2) and (4.17) is equivalent to

DfPtx - r(~-~ a) x = APtQx, x E D(A), t > O. (4.21)

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Taking x E X, a sequence {x n } C D(A) exists such that X n -+ x. Then from(4.21) and the boundedness of Df pta it follows

Now from the closedness of A we have Ptx E D(A) and y = APtax. Therefore(4.21) holds for every x E X.O

5. Examples

The next examples illustrate the theory in the preceding sections.

EXAMPLE 5.1. Consider the fractional diffusion problem (see [8], [13]):

Dfu = 82u/8x2, -00 < X < 00, t ~ 0, 0 < 0: < 1; u(=foo, t) = 0, u(x,O) = f(x).

Let X = L2(R), Af = f" with D(A) = W 2,2(R). Since A generates a holomor­phic semigroup, the conditions of Theorem 4.1. are satisfied and the solution hasthe properties (a)-(e). In this case the solution is given explicitly by (see [13]):

Similar results hold for a more general fractional diffusion problem in R n withAf = \72 f (see [20]).

EXAMPLE 5.2. Consider the problem

Dfu = -8u/8x, 0 < x < 1, t ~ 0, °< 0: < 1; u(O, t) = 0, u(x,O) = f(x).

Let X = L2(0, 1), Af = f' with D(A) = {f E Wl,2(0, 1), f(O) = O}. It is well­known that A generates an uniformly bounded Co-semigroup and using Proposi­tion 4.1. the properties (a)-(e) of the solution follow again. Moreover from (4.2)we obtain the following explicit representation of the solution:

EXAMPLE 5.3. If X = L2(0,1), A'of = _e iB f' with D(A'o) {f EW 1,2(0, 1),f(O) = O} we have the problem

Dfu = -eiB 8u/8x, 0 < x < 1, t ~ 0, 0 < 0: < 1; u(O, t) = 0, u(x, 0) = f(x).

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with solution (obtained by Laplace transform method)

u(x, t) = e-iOC cx fox <I>,;x(se-iOt-cx)j(x - s) ds

for 101 ~ O~, where O~ = (1 - ex)1r/2. Therefore Ae E Ccx iff 101 ~ O~, while Aegenerates a Co-semigroup only for 0 = 0. Taking 101 < O~, 0 =1= 0, Theorem 4.2.implies that the solution is even holomorphic although Ae does not generate aCo-semigroup.

EXAMPLE 5.4. If X = L 2(0, 1), A~j = ei01" with D(A~) = {f E W 2,2(0, 1),j(O) = j(l) = O} the corresponding problem is

Dfu = eiB f)2 u/f)x 2,°< x < 1, t 2: 0, 0< ex < 1; u(O, t) = u(l, t) = 0, u(x, 0) = j(x).

Since A~ has eigenvalues Zn = _eiB n21r2 and eigenfunctions sin n1rX, then if j(x) =L~=l Cn sin n1rX the solution is

00

u(x,t) = Lcn sinn1rxEcx (-e iB n21r2tcx).n=l

From the asymptotic expansion of the Mittag-Leffler function (2.7-8) it is clearthat A~ E Ccx iff 101 ~ O~, where O~ = (1- ex/2)1r. Moreover if IOJ < O~ the solutionis holomorphic. So A~ with 101 < O~, ReO < 0, is another example of an operatorgenerating holomorphic solution of (1.1-2) but not generating Co-semigroup (noright half-plane is free of spectra of A~).

Acknowledgments

The aut'or is grateful to Professor J. de Graaf for suggesting this problem andmany helpfuL discussions. Thanks are also due to the referee for drawing ourattention toward some relevant references.

References

[1] M. Cap u to, Linear models of dissipation whose Q is almost frequencyindependent, Part II. Geophys. J. R. Astr. Soc. 13 (1967),529-539.

[2] M. Cap u t 0, Elasticitri e Dissipazione. Zanichelli, Bologna (1969). (inItalian: Elasticity and Dissipation)

[3] M. Cap u t 0, F. M a ina r d i, Linear models of dissipation in anelasticsolids. Riv. Nuovo Cimento (Ser. II) 1 (1971),161-198.

[4] M.M. D j r bas h jail, Harmonic Analysis and Boundary Value Problemsin the Complex Plain. Ser. Oper. Theory: Adv. App!. 65, Birkhauser, Basel(1993).

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[5] A. E r del y i ( Ed.), Higher Transcendental Functions. McGraw-Hill,New-York (1955), Vol. 3, Ch. 18.

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[9] R. G 0 r e n flo, F. M a ina r d i, Integral and differential equations offractional order. in: A. Carpinteri, F. Mainardi (Eds.) , Fractals and Frac­tional Calculus in Continuum Mechanics. Springer Verlag, Wien and NewYork (1997),223-276. (Vol. no 378, series CISM Courses and Lecture Notes)

[10] A.N. K 0 c hub e i, A Cauchy problem for evolution equations of fractionalorder. J. Diff. Eqns 25 (1989), 967-974. (English transl. from Russian)

[ll] A.N. K 0 c hub e i, Fractional order diffusion. J. Diff. Eqns 26 (1990),485-492. (English transl. from Russian)

[12] V.A. K 0 s tin, The Cauchy problem for an abstract differential equationwith fractional derivatives. Russian Acad. Sci. Dokl. Math. 46 2 (1993),316-319. (English transl. from Russian)

[13] F. M a ina r d i, Fractional relaxation-oscillation and fractional diffusion­wave phenomena. Chaos, Solitons and Fractals 79 (1996), 1461-1477.

[14] F. M a ina r d i, Applications of Fractional Calculus in Mechanics. in: P.Rusev, I. Dimovski, V. Kiryakova (Eds.), Transform Methods and SpecialFunctions, Varna 1996. SCTP, Singapore (1997),309-334.

[15] F. M a ina r d i, M. Tom i rot t i, On a Special Function arising in thetime fractional diffusion-wave equation. in: P. Rusev, I. Dimovski, V.Kiryakova (Eds.) Transform Methods and Special Functions, Sofia 1994.SCTP, Singapore (1995), 171-183.

[16] K. S. Mille r, B. R 0 s s, An Introduction to the Fractional Calculus andFractional Differential Equations. Wiley, New York (1993).

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[18] J. P r ii s s, Evolutionary Integral Equations and Applications. Birkhauser,Basel, Boston, Berlin (1993).

[19] A.M.A. E I-S aye d, Fractional order evolution equations. Journal of Frac­tional Calculus 7 May (1995), 89-100.

[20] A.M.A. E I-S aye d, Fractional order diffusion-wave equation. InternationalJournal of Theoretical Physics 35 2 (1996),311-322.

[21] E.M. W rig h t, The generalized Bessel function of order greater than one.quarterly Journal of Mathematics (Oxford ser.) 11 (1940),36-48

[22] K. Yo sid a, Functional Analysis. (6th edition) Springer Verlag (1980).

*) Department of Mathematics and Computing Science,Eindhoven University of Technology,P. O. Box 513,5600 MB Eindhoven - The NETHERLANDS

e-mail: emilia@win.tue.nl

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Received: March, 1998