Martin Luther Universität Halle Wittenberg Institut für Mathematik

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1 Martin Luther Universität alle Wittenberg Institut für Mathematik Weak solutions of abstract evolutionary integro-ifferential equations in ilbert spaces Rico Zacher Report No. 1 28

2 Eitors: Professors of the Institute for Mathematics, Martin-Luther-University alle-wittenberg. Electronic version: see

3 Weak solutions of abstract evolutionary integro-ifferential equations in ilbert spaces Rico Zacher Report No Dr. Rico Zacher Martin-Luther-Universität alle-wittenberg Naturwissenschaftliche Fakultät III Institut für Mathematik Theoor-Lieser-Str. 5 D-612 alle/saale, Germany rico.zacher@mathematik.uni-halle.e

4 Weak solutions of abstract evolutionary integro-ifferential equations in ilbert spaces Rico Zacher aress: Martin-Luther University alle-wittenberg, Institute of Mathematics, Theoor-Lieser-Strasse 5, 612 alle, Germany, Abstract We prove existence an uniqueness of weak solutions to certain abstract evolutionary integro-ifferential equations in ilbert spaces, incluing evolution equations of fractional orer less than 1. Our results apply, e.g., to parabolic partial integro-ifferential equations in ivergence form with merely boune an measurable coefficients. AMS subject classification: 45N5, 45K5 Keywors: weak solution, integro-ifferential equation, fractional erivative, Galerkin metho 1 Introuction Let V an be real separable ilbert spaces such that V is ensely an continuously embee into. Ientifying with its ual we have V V, an h, v = h, v V V, h, v V, 1 where, an, V V enote the scalar prouct in an the uality pairing between V an V, respectively. In this paper we stuy the abstract problem [k u x]t, v t + at, ut, v = ft, v V V, v V, a.a. t, T, 2 where /t means the generalize erivative of real functions on, T, k L 1, loc R + is a scalar kernel that belongs to a certain kernel class k is of type PC, see Definition 2.1 below, k u stans for the convolution on the positive halfline, i.e. k ut = t kt τuτ τ, t, an a :, T V V R is a boune V-coercive bilinear form. Further, x an f L 2 [, T ]; V are given ata. We seek a solution u of 2 in the regularity class W x, V, := {u L 2 [, T ]; V : k u x 1 2 [, T ]; V }, where the zero means vanishing trace at t =. The vector x can be regare as initial ata for u, at least in a weak sense. If e.g. u, an t k [u x] belong to C[, T ]; V, then the conition k u x = implies u = x, see Section 3. Work partially supporte by the Deutsche Forschungsgemeinschaft DFG, Bonn, Germany. 1

5 An important example for the kernel k we have in min is given by where g β enotes the Riemann-Liouville kernel kt = g 1 α te µt, t >, α, 1, µ, 3 g β t = tβ 1, t >, β >. Γβ In this case, 2 amounts to an abstract ifferential equation of fractional orer α, 1. Recall that for a sufficiently smooth function v on R +, the Riemann-Liouville fractional erivative D β t v of orer β, 1 is efine by D β t v = t g 1 β v. In this paper we prove that the problem 2 possesses exactly one solution in the class W x, V,. This result can be regare as the analogue of the well-known existence an uniqueness result for the corresponing abstract parabolic equation t ut, v + at, ut, v = ft, v V V, v V, a.a. t, T, u = x, u 1 2 [, T ]; V L 2 [, T ]; V, see e.g. Theorem 4.1 an Remark 4.3 in Chapter 4 in [7] or [12, Section 23]. We point out that concerning time regularity the bilinear form a is only assume to be measurable in t. This allows, e.g., to treat parabolic partial integro-ifferential equations in ivergence form with merely boune an measurable coefficients, see Section 4. The proof of the main result is base on the Galerkin metho an suitable a priori estimates for weak solutions of 2. These estimates are erive by means of the basic ientity 17 see below for absolutely continuous kernels. It has been known before but oes not seem to appear in the literature in the context of problems of the form 2. We remark that recently [9] the ientity 17 was successfully employe to construct Lyapunov functions for certain nonlinear ifferential equations of fractional orer between an 2. In orer to be able to apply 17, we approximate the kernel k by the sequence k n which is obtaine from the Yosia approximation of the operator B efine by Bv = t k v, e.g. in L 2 [, T ]. This metho was alreay use in [9], we also refer to [5], where a more general class of integro-ifferential operators in time is stuie. Note that 2 is equivalent to the equation [k u x]t + Atut = ft, a.a. t, T, 5 t in V, where the operator At : V V is efine by Atu, v V V = at, u, v, u, v V. 6 For equations of the form 5 with At A there exists a vast literature, even in general Banach spaces, see e.g. [5], an [8] an the references given therein. owever, in the case of time-epenent A an without smoothness assumption nothing seems to be known in the literature concerning existence an uniqueness. The paper is organize as follows. In Section 2 we introuce the notion of kernels of type PC, an escribe the approximation of such kernels use in this paper. We further prove a 4 2

6 trace theorem for functions in the class W x, V,, an state the basic ientity 17. Section 3 contains the main result on existence an uniqueness as well as some further interpolation results for functions from W x, V,. We also look at the special case of fractional evolution equations. In Section 4 we apply the abstract results to secon-orer parabolic partial integro-ifferential equations in ivergence form. 2 Preliminaries The following class of kernels is basic to our treatment of 2. Definition 2.1 A kernel k L 1, loc R + is calle to be of type PC if it is nonnegative an nonincreasing, an there exists a kernel l L 1, loc R + such that k l = 1 in,. In this case, we say that k, l is a PC pair an write k, l PC. From k, l PC it follows that l is completely positive see e.g. Theorem 2.2 in [2], in particular l is nonnegative, cf. [2, Proposition 2.1]. An important example is given by kt = g 1 α te µt an lt = g α te µt + µ1 [g α e µ ]t, t >, 7 with α, 1 an µ. Both k an l are strictly positive an ecreasing; observe that l t = g αte µt <, t >. The Laplace transforms are given by ˆkλ = 1 λ + µ 1 α, 1 ˆlλ = 1 λ + µ α + µ, Re λ >, λ which shows that k l = 1 on,. ence we have both k, l PC, an l, k PC. PC pairs enjoy a useful stability property with respect to exponential shifts. Writing k µ t = kte µt, t >, µ, we have k, l PC k µ, l µ + µ1 l µ PC, µ. 8 To prove 8, we first note that for any µ, k µ is eviently nonnegative an nonincreasing, an l µ + µ1 l µ is nonnegative. Multiplying k l = 1 by 1 µ t = e µt gives k µ l µ = 1 µ, which in turn implies that µk µ 1 l µ = µ1 1 µ = 1 1 µ. Aing these relations, we see that k µ [l µ + µ1 l µ ] = 1. We next iscuss an important metho of approximating kernels of type PC. Let k, l PC, T >, an be a real ilbert space. Then the operator B efine by Bu = t k u, DB = {u L 2[, T ]; : k u 1 2 [, T ]; }, is known to be m-accretive in L 2 [, T ];, cf. [1], [3], an [5]. Its Yosia approximations B n, efine by B n = nbn + B 1, n N, enjoy the property that for any u DB, one has B n u Bu in L 2 [, T ]; as n. Furthermore, we have the representation B n u = t k n u, u L 2 [, T ];, n N, 9 3

7 where k n = ns n, with s n being the solution of the scalar-value Volterra equation s n t + nl s n t = 1, t >, n N, see e.g. [9]. The kernels s n, n N, are nonnegative an nonincreasing in,, an s n 1 1 [, T ], cf. [8, Prop. 4.5]. Consequently, the kernels k n, n N, enjoy the same properties. Moreover, k n k in L 1 [, T ] as n. Let now V an be real ilbert spaces as escribe above, that is V V. In the theory of abstract parabolic equations the continuous embeing 1 2 [, T ]; V L 2 [, T ]; V C[, T ]; 1 is well-known, see e.g. Proposition 2.1 an Theorem 3.1 in Chapter 1 of [7], or Proposition in [12]. The following theorem provies the analogue of 1 in the case of the space W x, V,. Theorem 2.1 Let V an be real ilbert spaces as escribe above V V. Let further T >, an k L 1, loc R + be of type PC. Suppose that x, an u W x, V,. Then k u x an k u belong to the space C[, T ]; after possibly being reefine on a set of measure zero. The mapping {t k u 2 t} is absolutely continuous on [, T ], with t k u 2 t = 2 [k u x] t, [k u]t] V V + 2kt x, k ut 11 for a.a. t [, T ]. Furthermore, k u C[,T ]; C t [k u x] L2[,T + u L 2[,T ];V + x, 12 ];V the constant C epening only on T, k L1[,T ], an the constant of the embeing V. We remark that in the case x =, the property k u C[, T ]; follows immeiately from the embeing 1. In fact, u L 2 [, T ]; V implies k u L 2 [, T ]; V, by Young s inequality, an so k u 1 2 [, T ]; V L 2 [, T ]; V C[, T ];. We point out that for x this simple reuction is not feasible. Proof of Theorem 2.1. Note first that k x = 1 k x 1 1 [, T ]; C[, T ];. Thus k u x C[, T ]; if an only if k u C[, T ];. Let k n 1 1 [, T ], n N, be the kernel associate with the Yosia approximation B n of the operator Bv = t k v, DB = {v L 2[, T ]; V : k v 1 2 [, T ]; V }. 13 Then k n u 1 2 [, T ]; V, an we have for n, m N, k n ut k m ut 2 [k t = 2 n u] t [k m u] t, [k n u]t [k m u]t. 4

8 Thus, in view of 1 an Young s inequality, k n ut k m ut 2 = k n us k m us t s t s [k n u x] τ [k m u x] τ, [k n u]τ [k m u]τ τ V V [k n τ k m τ] x, [k n u]τ [k m u]τ k n us k m us 2 + t [k n u x] t [k m u x] + k n u k m u x 2 k n k m 2 L L 2[,T ];V 1[,T ] + 1 k n u k m u 2 τ 2 2 L 2[,T ];V 2 C[,T ]; 14 for all s, t [, T ]. Since k n k in L 1 [, T ] as n, we have k n u k u in L 2 [, T ]; as well as in L 2 [, T ]; V. Further, u x DB implies that t [k n u x] t [k u x] in L 2 [, T ]; V. We fix now a point s, T for which k n us k us in as n. Taking then in 14 the maximum over all t [, T ] an absorbing the last term, it follows that k n u is a Cauchy sequence in C[, T ];. Thus k n u converges in C[, T ]; to some v C[, T ];. Since we also know that k n u k u in L 2 [, T ];, we euce k u = v a.e. in [, T ], proving the first part of the theorem. Similarly as above we see that k n ut 2 = k n us t s t s [k n u x] τ, [k n u]τ k n τ x, k n uτ τ for all s, t [, T ], an n N. Taking the limits as n we obtain k ut 2 = k us t s t s [k u x] τ, [k u]τ kτ x, k uτ V V τ V V τ τ 15 for all s, t [, T ]. ence {t k u 2 t} is absolutely continuous on [, T ], an 11 hols true. To obtain 12, we estimate the integral terms in 15 similarly as for 14 an integrate with respect to s. This yiels k ut 2 1 T k u 2 L 2[,T ]; + t [k u x] 2 L 2[,T ];V + k u 2 L 2[,T ];V + 2 x 2 k 2 L 1[,T ] k u 2 C[,T ]; 16 5

9 for all t [, T ]. We then take the maximum over all t [, T ], absorb the last term, an use Young s inequality for convolutions, to the result 1 2 k u 2 C[,T ]; 1 T k 2 L 1[,T ] u 2 L 2[,T ]; + t [k u x] 2 + k 2 L 1[,T ] u 2 L 2[,T ];V + 2 x 2 k 2 L 1[,T ], L 2[,T ];V which implies 12. The following lemma is of funamental importance with regar to a priori estimates for 2 an certain interpolation results for functions in the class W x, V,. Lemma 2.1 Let be a real ilbert space an T >. Then for any k 1 1 [, T ] an any v L 2 [, T ]; there hols t k vt, vt = 1 2 t k v 2 t kt vt t [ ks] vt vt s 2 s, a.a. t, T. 17 The assertion of Lemma 2.1 follows from a straightforwar computation. We remark that a more general version of 17 with = R n in integrate form can be foun in [6, Lemma ]. 3 The main existence an uniqueness result In this section we are concerne with existence an uniqueness for the abstract problem 2. Recall that V an are real separable ilbert spaces such that V is ensely an continuously embee into. Ientifying with its ual, we have V V, an the relation 1 hols. It will be assume that im V =. We will suppose that the following assumptions are satisfie. k k, l PC for some l L 1, loc R +. x, f L 2 [, T ]; V. a For a.a. t, T, the mapping at,, : V V R is bilinear, an there exist constants M >, c >, an, which are inepenent of t, such that at, u, v M u V v V, 18 at, u, u c u 2 V u 2, 19 for all u, v V an a.a. t, T. Moreover, the function {t at, u, v} is measurable on, T for all u, v V. We seek a solution of 2 in the space W x, V, = {u L 2 [, T ]; V : k u x 1 2 [, T ]; V }. 6

10 Note that the vector x plays the role of the initial ata for u, at least in a weak sense. If e.g. u, an t k [u x] =: f belong to C[, T ]; V, then the assumption k an the conition k u x = entail that u x = l k [u x] = l f t in C[, T ]; V, an therefore u = x. In orer to construct a solution in the esire class, we will use the Galerkin metho. We will assume that b {w 1, w 2,...} is a basis in V, an x m is a sequence in such that x m span{w 1,..., w m }, m N, an x m x in as m. Setting m m u m t = c jm tw j, x m = β jm w j, j=1 an replacing u, x, an v in 2 by u m, x m, an w i, respectively, we formally obtain for every m N, the system of Galerkin equations m j=1 t [k c jm β jm ]tw j, w i + j=1 m c jm tat, w j, w i = ft, w i V V, 2 j=1 for a.a. t, T, where i runs through the set {1,..., m}. The main result in this section is the following. Theorem 3.1 Let T >, an V an be real ilbert spaces as escribe above. Suppose the assumptions k,, a, an b hol. Then the problem 2 has exactly one solution u in the space W x, V,. The mapping x, f u is linear, an there exists a constant M > such that k u x 1 2 [,T ];V + u L2[,T ];V M x + f L2[,T ];V 21 for all x an f L 2 [, T ]; V. Moreover, for every m N, the Galerkin equation 2 possesses precisely one solution u m W x m, V,. The sequence u m converges weakly to u in L 2 [, T ]; V as m. Proof. Uniqueness. Suppose that u 1, u 2 W x, V, are solutions of 2. The ifference u = u 1 u 2 then belongs to the space W, V, an satisfies the equation k u t, v V V + at, ut, v =, v V, a.a. t, T. We may take v = ut, thereby getting k u t, ut V V + at, ut, ut =, a.a. t, T. 22 Let k n 1 1 [, T ], n N, be the kernel associate with the Yosia approximation B n of the operator B efine in 13. Then 22 is equivalent to k n u t, ut V V + at, ut, ut = h n t, a.a. t, T, 23 7

11 where h n t = k n u t k u t, ut V V, a.a. t, T. Since k n u 1 2 [, T ];, we may apply 1 to the first term in 23, to the result t k n ut, ut + at, ut, ut = h n t, a.a. t, T, 24 for all n N. The kernels k n are nonnegative an nonincreasing. Thus, by Lemma 2.1, 1 2 t k n u 2 t t k n ut, ut, a.a. t, T. The secon term in 24 is estimate by means of the abstract Gåring inequality 19 in a. Proceeing this way, it follows from 24 that t k n u 2 t 2 ut 2 + 2h n t, a.a. t, T. 25 Observe that all terms in 25 viewe as functions of t belong to L 1 [, T ]. Therefore we may convolve 25 with the kernel l from assumption k. Letting then n go to an selecting an appropriate subsequence, if necessary, we arrive at ut 2 2 l u 2 t, a.a. t, T. 26 ere we use the fact that h n in L 1 [, T ], which entails l h n in L 1 [, T ], an that l t k n u 2 = t k n l u 2 t k l u 2 = u 2 in L 1 [, T ] as n. Since l is nonnegative, 26 implies that ut 2 = a.e. in, T, by the abstract Gronwall lemma [11, Prop. 7.15], i.e. u =. Existence. 1. We show first that for every m N, the system of Galerkin equations 2 amits a unique solution ψ := ψ m := c 1m,..., c mm T on [, T ] in the class W ξ, R m, R m, where ξ := ξ m := β 1m,..., β mm T. Since the vectors w 1,..., w m are linearly inepenent, the matrix w j, w i R m m is invertible. ence 2 can be solve for t [k c jm β jm ], which leas to an equivalent system of the form [k ψ ξ]t = Btψt + gt, a.a. t, T, 27 t where B L [, T ]; R m m, an g L 2 [, T ]; R m, by the assumptions a an. In orer to solve 27, we transform it into the system of Volterra equations ψt = ξ + l [B ψ ]t + l gt, a.a. t, T, which has a unique solution ψ L 2 [, T ]; R m, see e.g. [6, Chapter 9]. But then ψ W ξ, R m, R m, an hence it is also a solution of 27. This shows that for every m N, the Galerkin equation 2 has exactly one solution u m W x m, V,. 8

12 2. We next erive a priori estimates for the Galerkin solutions. The Galerkin equations 2 are equivalent to t [k u m x m ]t, w i + at, u mt, w i = ft, w i V V, a.a. t, T, 28 i = 1,..., m. Multiplying 28 by c im an summing over i, we obtain t [k u m x m ]t, u m t + at, u m t, u m t = ft, u m t V V. 29 Let k n 1 1 [, T ], n N, be as in the uniqueness part above. Then 29 can be written as with t k n u m t, u m t + at, u m t, u m t = k n tx m, u m t + ft, u m t V V + h mn t, a.a. t, T, 3 h mn t = [k n u m x m ] t [k u m x m ] t, u m t V V. Using Lemma 2.1 an inequality 19, we fin that 1 2 t k n u m 2 t k nt u m t 2 + c u m t 2 V u m t 2 + k n tx m, u m t + ft, u m t V V + h mn t, which, by Young s inequality, yiels the estimate t k n u m 2 t + c u m t 2 V 2 u m t 2 + k n t x m c ft 2 V + 2h mnt. 31 Similarly as in the uniqueness part, we see that l h mn an l t k n u m 2 u m 2 in L 1 [, T ] as n. Consequently, if we convolve 31 with l, an let n ten to, selecting an appropriate subsequence, if necessary, we obtain the estimate u m t 2 2 l u m 2 t + x m c l f 2 V t 32 for a.a. t, T, an all m N. By positivity of l, it follows from 32 that u m L2[,T ]; C x m + f L2[,T ];V, 33 where the constant C epens only on c,, l, T. Returning to 31, we may integrate from to T - note that k n u m 2 = - an then let n go to to fin that c u m t 2 V t 2 u m t 2 t + k L1[,T ] x m ft 2 V c t. 9

13 This, together with 33 an the assumption x m x in, yiels the a priori boun u m L2[,T ];V C 1 x + f L2[,T ];V, m N, 34 with some C 1 > being inepenent of m N. 3. By 34 there exists a subsequence of u m, which we will again enote by u m, such that u m u in L 2 [, T ]; V as m, 35 for some u L 2 [, T ]; V. We will show that u W x, V,, an that u is a solution of 2. Let ϕ C 1 [, T ]; R with ϕt =. Multiplying 28 by ϕ an using integration by parts, we obtain ϕ t[k u m x m ]t, w i t + ϕtat, u m t, w i t = ϕt ft, w i V V 36 for all m i, because [k u m x m ] =. We apply then the limits 35, an x m x in to equation 36. By means of 18, the embeing V, an Young s an öler s inequality, one easily verifies that this leas to ϕ t[k u x]t, w i t + ϕtat, ut, w i t = ϕt ft, w i V V 37 for all i N. Observe that [k u x]t, w i = [k u x]t, w i V V, by 1. It is not ifficult to see that the terms in 37 represent linear continuous functionals on the space V, with respect to w i. Consequently, in light of b, 37 implies ϕ t [k u x]t, v V V t + ϕtat, ut, v t = ϕt ft, v V V 38 for all v V. Since 38 hols in particular for all ϕ C, T, we infer that k u x has a generalize erivative on, T with [k u x]t + Atut = ft, a.a. t, T, 39 t where the operator At : V V is efine as in 6. From u L 2 [, T ]; V an Atut V M ut V for a.a. t, T, we euce that A u L 2 [, T ]; V. Since f L 2 [, T ]; V, too, it follows that [k u x] L 2 [, T ]; V. To see that u W x, V,, it remains to show that [k u x] =. We set z := k u x. Then z 2 1 [, T ]; V C[, T ]; V, an by 38 an 39, there hols ϕ t zt, v V V t = ϕt z t, v V V 4 1

14 for all v V, an all ϕ C 1 [, T ]; R with ϕt =. Choosing ϕ such that ϕ = 1, an approximating z in 2 1 [, T ]; V by a sequence of functions z n C 1 [, T ]; V, it follows from 4 an the formula of integration by parts that z, v V V = for all v V. ence z =. Summarizing, we have foun a function u W x, V, that solves the operator equation 39. Since 39 is equivalent to 2, the existence proof is complete. Moreover, 39 has exactly one solution in the class W x, V,. Consequently, all subsequences of the original sequence u m that are weakly convergent in L 2 [, T ]; V have the same limit u. ence, the original sequence u m converges weakly to u in L 2 [, T ]; V. Continuous epenence on the ata. From u m u in L 2 [, T ]; V an the estimate 34, it follows by means of the theorem of Banach an Steinhaus, that u L2[,T ];V lim inf u m L2[,T ];V C 1 x + f L2[,T ];V m. Using this estimate, together with A u L2[,T ];V M u L2[,T ];V,, an 39, we obtain the esire estimate 21. If not only the kernel l in k but also some p-th power of it with p > 1 belongs to L 1 [, T ], then one can get an aitional estimate for solutions of 2. This is the consequence of the first part of the subsequent interpolation result for functions in the space W x, V,. It also contains the analogue of t 1 ut 2 t < for all u 1 2 [, T ]; V L 2 [, T ]; V, see [7, Chap. 3, Prop. 5.3 an Prop. 5.4], in the case of the space W x, V,. By L p,w [, T ], p [1, we mean the weak L p space of Lebesgue measurable functions on, T. Theorem 3.2 Let V an be real ilbert spaces as escribe above V V. Let further T >, k, l PC, an suppose that x, an u W x, V,. Then the following statements hol. i If l L p,w [, T ] for some p > 1 then u L 2p,w [, T ];, an there hols u L2p,w[,T ]; C t [k u x] L2[,T + u L 2[,T ];V + x, 41 ];V the constant C epening only on T, an l Lp,w[,T ]. If l L p [, T ] for some p > 1 then u L 2p [, T ];, an the estimate corresponing to 41 hols. ii There hols the estimate kt ut 2 t 1/2 C 1 t [k u x] L2[,T + u L 2[,T ];V + x, ];V where the constant C 1 only epens on k L1[,T ]. 11

15 Proof. We procee similarly as in the proof of the previous result. The key iea again is to apply the ientity 17 from Lemma 2.1. Let k n be the sequence of approximating kernels use above, an put an gt = k [u x] t, ut V V, t, T, h n t = k n [u x] t k [u x] t, ut V V, t, T. Then t k n ut, ut = k n tx, ut + gt + h n t, a.a. t, T, with each term being in L 1 [, T ]. Using 17 an the inequality ab 1 4 a2 + b 2 it follows that 1 2 t k n u 2 t k nt ut 2 k n t x 2 + gt + h n t, a.a. t, T. 42 To prove i, we rop the secon term on the left, which is nonnegative, convolve the resulting inequality with l, an sen n to. Arguing as in the proof of Theorem 3.1 we obtain ut 2 2 x 2 + l gt, a.a. t, T. Young s inequality for weak type L p spaces see e.g. [4, Theorem ] then gives u 2 L 2p,w[,T ]; u = 2 l Lp,w[,T 2 Lp,w[,T ] g L1[,T ] + T 1/p x 2, ] which together with 2 g L1[,T ] t [k u x] 2 L 2[,T ];V + u 2 L 2[,T ];V implies the first esire boun in i. If l L p [, T ] one may apply Young s classical inequality for convolutions to establish the asserte estimate in i. As to ii, we integrate 42 from to T, an rop the term k n u 2 T, to the result k n t ut 2 t 4 1 k n T x 2 + g L1[,T ] + h n L1[,T ]. Observe that 1 k n T 1 kt, an h n L1[,T ] as n. Therefore, for sufficiently large n we have k n t ut 2 t 8 1 kt x 2 + g L1[,T ]. Since k n k in L 1 [, 1], the assertion then follows from Fatou s lemma. In the case of fractional evolution equations we have the following corollary. ere we set α 2 [, T ]; V := {v [,T ] : v α 2 R; V an supp v R + }, where α 2 R; V stans for the Bessel potential space of orer α of V -value functions on the line. 12

16 Corollary 3.1 Let T >, an V an be real ilbert spaces as escribe above. Suppose, a, an b, an assume that kt = g 1 α te µt, t >, with α, 1, an µ. Then the problem 2 amits exactly one solution u in the space Furthermore, we have W α; x, V, := {u L 2 [, T ]; V : u x α 2 [, T ]; V }. g 1 α e µ u C[, T ];, u L 2 1 α, w [, T ];, an t α ut 2 t <. There exists a constant M = Mα, µ, T > such that u x 2 α[,t ];V + u L2[,T ];V+ g 1 α e µ u C[,T ]; + u L 2, w[,t ]; 1 α + t α ut 2 t 1/2 M x + f L2[,T ];V, for all x an f L 2 [, T ]; V. Proof. Let l as in 7. Then α 2 [, T ]; V = {l v : v L 2 [, T ]; V } = {v L 2 [, T ]; V : g 1 α e µ v 1 2 [, T ]; V, for the first equals sign see e.g. [1, Corollary 2.1]; the secon one follows from k l = 1. Note further that l L 1/1 α,w [, T ]. So the assertions of the corollary follow immeiately from the previous results. Note that taking formally the limit α 1 in the above estimates with µ = we recover the well-known estimates for solutions of the abstract parabolic equation 4. 4 Example Let T >, an Ω be a boune omain in R N with N 3. We consier the problem t k u u iv A Du + b Du + cu = g, t, T, x Ω, ut, x =, t, T, x Ω, u, x = u x, x Ω. 43 ere Du enotes the graient of u w.r.t. the spatial variables, an z 1 z 2 means the scalar prouct of z 1, z 2 R N. We make the following assumptions on the kernel k, the coefficients, an the ata. k k, l PC for some l L 1, loc R +. u L 2 Ω, g L 2 [, T ]; L 2N Ω. N+2 13

17 A A L, T Ω; R N N, an ν > such that At, xξ ξ ν ξ 2, for a.a. t, T, x Ω, an all ξ R n. c b L, T Ω; R N, c L, T Ω. We set V = 2 1 Ω, an = L 2 Ω, enowe with the inner prouct u, v = uv x. Define Ω at, u, v = At, xdux Dvx + bt, x Duxvx + ct, xuxvx x an Ω ft, v V V = Then the weak formulation of 43 reas Ω gt, xvx x, a.a. t, T. [k u u ]t, v t + at, ut, v = ft, v V V, v V, a.a. t, T, 44 an we seek a solution in the class W u, 1 2 Ω, L 2 Ω={u L 2 [, T ]; 1 2 Ω : k u u 1 2 [, T ]; 1 2 Ω}. It is folklore that in the escribe setting the assumptions, an a in Theorem 3.1 are satisfie. Concerning b, we coul take {w 1, w 2,...} to be the complete set of eigenfunctions for in 1 2 Ω. Consequently, we obtain Corollary 4.1 Suppose the assumptions k,, A, an c hol. Then the problem 43 has a unique weak solution u W u, 1 2 Ω, L 2 Ω in the sense that 44 is satisfie. Further, k u C[, T ]; L 2 Ω. In the case kt = g 1 α te µt, t >, with α, 1, an µ, we have u L 2 1 α, w [, T ]; L 2 Ω L 2 [, T ]; 1 2 Ω, an u u α 2 [, T ]; 1 2 Ω}. Of course, a corresponing result also hols in the case N 2 with the assumption on f appropriately moifie. References [1] Clément, Ph.: On abstract Volterra equations in Banach spaces with completely positive kernels. Infiniteimensional systems Retzhof, 1983, pp. 32-4, Lecture Notes in Math., 176, Springer, Berlin, [2] Clément, Ph.; Nohel, J.A.: Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM J. Math. Anal , pp [3] Clément, Ph.; Prüss, J.: Completely positive measures an Feller semigroups. Math. Ann , pp [4] Grafakos, L.: Classical an moern Fourier analysis. Pearson Eucation Inc., Upper Sale River, New Jersey 7458,

18 [5] Gripenberg, G.: Volterra integro-ifferential equations with accretive nonlinearity. J. Differ. Eq , pp [6] Gripenberg, G.; Lonen, S.-O.; Staffans, O.: Volterra integral an functional equations. Encyclopeia of Mathematics an its Applications, 34. Cambrige University Press, Cambrige, 199. [7] Lions, J. L.; Magenes, E.: Non-homogeneous bounary value problems an Applications. Volume I, Springer, Berlin, [8] Prüss, J.: Evolutionary Integral Equations an Applications. Monographs in Mathematics 87, Birkhäuser, Basel, [9] Vergara, V.; Zacher, R.: Lyapunov functions an convergence to steay state for ifferential equations of fractional orer. Preprint 27. To appear in Math. Z. [1] Zacher, R.: Maximal regularity of type L p for abstract parabolic Volterra equations. J. Evol. Equ. 5 25, pp [11] Zeiler, E.: Nonlinear functional analysis an its applications. I: Fixe-point theorems. Springer-Verlag, New York, [12] Zeiler, E.: Nonlinear functional analysis an its applications. II/A: Linear monotone operators. Springer- Verlag, New York,