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Annals of the Alexandru Ioan Cuza University - Mathematics
, Volume 61 (1) – Jan 1, 2015

/lp/de-gruyter/identification-of-a-source-term-in-a-semilinear-evolution-delay-iCFYmnRGQb

- Publisher
- de Gruyter
- Copyright
- Copyright © 2015 by the
- ISSN
- 1221-8421
- eISSN
- 1221-8421
- DOI
- 10.2478/aicu-2013-0003
- Publisher site
- See Article on Publisher Site

An existence, uniqueness and continuous dependence on the data result for a source term identification problem in a semilinear functional delay differential equation in a general Banach space is established. As additional condition, it is assumed thahe mean of the solution, with respeco a non-atomic Borel measure, is a preassigned element in the domain of the linear part of the right-hand side of the equation. Two applications to source identification, one in a parabolic functional delay equation and another one in a hyperbolic delay equation, are also discussed. Mathematics Subject Classification 2010: 34K29, 34K30, 35K58, 35L02, 47D06, 47H10. Key wor: identification problem, first-order semilinear functional differential delay equation, unknown source, C0 -semigroup of contractions, parabolic functional delay equation, first-order hyperbolic functional delay equation. 1. Introduction Let X be a Banach space whose norm is denoted by · , let A : D(A) X X be the infinitesimal generator of a C0 -semigroup of contractions {S(t); t 0}, let (0, 1), let : [ 0, 1 ] × C([ -, 0 ]; X) R be a nonlinear C 1 -functional, let f : [ 0, 1 ]×C([ -, 0 ]; X) X be a C 1 -function and let W 1,1 ([ 0, 1 ]; R) \ {0} be a positive function. Throughout, if The first author is a member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica. Note. Regretfully, on 9 November 2013, the first author, Professor Alfredo Lorenzi, passed away suddenly. ALFREDO LORENZI and IOAN I. VRABIE u C([ -, 1 ]; X) and t [ 0, 1 ], ut denotes the function in C([ -, 0 ]; X), defined by ut (s) := u(t + s) for each s [ -, 0 ]. In this paper we prove an existence, uniqueness and continuous dependence on the data result for a source term identification problem: (IP) given h C 1 ([ -, 0 ]; X) and X, find z X and a mild solution u : [ 0, 1 ] X of the Cauchy Problem (1.1) u (t) = Au(t) + (t, ut )z + f (t, ut ), u(t) = h(t), t [ 0, 1 ], t [ -, 0 ], satisfying the additional condition (1.2) (t)u(t) dt = . The constant delayed case, i.e. (1.3) u (t)=Au(t)+(t, u(t), u(t- ))z+f (t, u(t), u(t- )), u(t)=h(t), t [0, 1], t [-, 0], where : [ 0, 1 ] × X × X R and f : [ 0, 1 ] × X × X X, was considered in Lorenzi-Vrabie [9]. Since (t, u(t), u(t - )) = (t, (s - 0)ut , (s - )ut ) = (t, ut ), f (t, u(t), u(t - )) = f (t, (s - 0)ut , (s - )ut ) = f (t, ut ), (s + a) being the Dirac measure concentrated at a, (1.3) is a particular case of (1.1). Previous results on the non-delayed linear versions of (1.3) was analyzed by Lorenzi and Vrabie [6], while the semilinear non-delayed case with depending only on u, (t) = 1 for t [ 0, 1 ], and f independent of ut , was considered in Lorenzi and Vrabie [7]. The semilinear non-delayed case with depending on t and being arbitrary in W 1,1 was analyzed in Lorenzi and Vrabie [8]. Identification problems, aimed at recovering a time-depending function in delay functional differential equations were studied by Di Blasio and Lorenzi [3] and [5], in the case of an unknown factor in the source, and in [4], in the case of an unknown convolution kernel. IDENTIFICATION OF A SOURCE It should be emphasized that, unlike in Lorenzi and Vrabie [7], where A is assumed to generate a compact C0 -semigroup which is an abstract parabolicity condition, here following the same strategy as in Lorenzi and Vrabie [9] we merely assume that (I - A) is compact, condition which is satisfied in both parabolic and hyperbolic problems. We recall that, if the C0 -semigroup generated by A is compact, then (I - A) is compact, but not conversely. See for instance Pazy [11, Theorem 6.2.1] or Vrabie [14, p. 134]. Definition 1.1. By definition, a strict solution of the equation (1.1) is a function u C([-, 1]; X) C 1 ([ 0, 1 ]; X) C([ 0, 1 ]; D(A))1 , satisfying u (t) = Au(t) + (t, ut )z + f (t, ut ) for each t [ 0, 1 ] as well as u(t) = h(t) for t [ -, 0 ]. Accordingly, by a strict solution of (IP) we mean a pair (z, u) such that u is a strict solution of the Cauchy problem (1.1) and, in addition, u verifies (1.2). The paper is divided into seven sections. Section 2 is devoted to some preliminaries while, in Section 3, we state our main existence result concerning mild solutions, i.e. Theorem 3.1. Section 4 is concerned with the proof of Theorem 3.1, Section 5 deals with a uniqueness and continuous dependence result, i.e., Theorem 6.1. In Sections 6 and 7 we include two examples interesting by themselves. The first one refers to an identification problem for a parabolic semilinear delay equation, while the latter is devoted to an identification problem for a hyperbolic delay problem. 2. Preliminaries 2.1. The compactness argument Let X be fixed, let F L1 (0, 1; X), g F and let us consider the evolution equation (2.1) u (t) = Au(t) + g(t), u(0) = . t [ 0, 1 ], Hereafter C([ 0, 1 ]; D(A)) is the space of all continuous functions from [ 0, 1 ] to D(A), the latter being endowed with the usual graph norm · A , defined by u A = u + Au for each u D(A). ALFREDO LORENZI and IOAN I. VRABIE Throughout X is assumed to be fixed and we denote by ug the unique mild solution of (2.1) corresponding to g F, i.e., ug (t) = S(t) + S(t - s)g(s) for t [ 0, 1 ], and by M(F) = {ug ; g F}, the set of mild solutions of (2.1) corresponding to all g F. A set F L1 (0, 1; X) is called Lebesgue-uniformly integrable if for each > here exists () > 0 such that, for each measurable subset E [ 0, 1 ] with Lebesgue measure (E) (), we have g(t) dt uniformly for g F. The next compactness result, for A being a nonlinear and m-dissipative operator and F being bounded in C([ 0, 1 ]; X), was proved by Vrabie [12]. The extension to the more general case when F is uniformly integrable in L1 (0, 1; X) was obtained by Mitidieri and Vrabie [10]. We confine ourselves to state below only the linear variant which is exactly what we need for our later purposes. For details see Vrabie [14, Theorem 8.5.1, p. 197]. Theorem 2.1. Let A : D(A) X X be the infinitesimal generator of a C0 -semigroup {S(t); t 0} and let us assume that (I - A) is compact. Let F L1 (0, 1; X) be a Lebesgue-uniformly integrable set and let X be fixed. Then, the following conditions are equivalent: (c1 ) the set M(F) is equicontinuous from the right on [ 0, 1) ; (c2 ) the set M(F) is relatively compact in C([ 0, 1 ]; X). 2.2. An equivalent problem We reformulate the identification problem (IP) as an equivalent nonlinear delay evolution equation more convenient for our later purposes. Let us assume for the time being that (z, u) is a strict solution of (IP). Clearly (t)u (t) dt = (1)u(1) - (0)h(0) - (t)u(t) dt. IDENTIFICATION OF A SOURCE Since A is closed, from Vrabie [14, Hille's Theorem 1.2.2, p. 8], we have (t)Au(t) dt = A (t)u(t) dt = A. Multiplying both sides of the equation in (1.1) by (t), integrating over [ 0, 1 ], taking into accounhe equalities above, we deduce (1)u(1) - (0)h(0) - (t)u(t) dt z + A + (t)f (t, ut ) dt. Let us assume further that, for each v C([ -, 0 ]; X), we have (2.2) (t)(t, v) dt m > 0, where m is independent of v. Clearly, in this case, we get (2.3) m > 0, for each u C([ -, 1 ]; X), (2.2) and (2.3) being equivalent. Under the circumstances mentioned above, we observe that (2.4) z = F (u), where F : C([ -, 1 ]; X) X (here h and are fixed) is defined by (2.5) F (u) = J(u) + (1)u(1) - (t)f (t, ut ) dt , X and J : C([ -, 1 ]; X) R being given by = -A - (0)h(0) 1 (2.6) J(u) := Whence, if (z, u) is a strict solution of (IP), then u is a strict solution of the problem (2.7) u (t) = Au(t) + (t, ut )F (u) + f (t, ut ), t [ 0, 1 ] u(t) = h(t), t [ -, 0 ], ALFREDO LORENZI and IOAN I. VRABIE F (u) being defined by (2.5). Under some additional easy to verify conditions, the converse statement hol also true. Namely, we have: Lemma 2.1. Let (2.2) be satisfied, let A be injective and let u be a strict solution of the problem (2.7), where F (u) is defined by (2.5), while and J(u) are defined by (2.6). Then u satisfies the additional condition (1.2), and so (F (u), u) is a strict solution of (IP). Since the proof of Lemma 2.1 follows the very same arguments as in the proof of Lemma 2.1 in Lorenzi and Vrabie [9], we do not enter into details. Following Lorenzi and Vrabie [9], we introduce two concepts of solution for our identification problem. We start with: Definition 2.1. By a generalized solution of the identification problem (IP) we mean a function u C([-, 1]; X) satisfying u(t) = h(t) for t [ -, 0 ] and u(t) = h(0) + S(t - s)(s, us )F (u) + S(t - s)f (s, us ) for t [ 0, 1 ], where F (u) is given by (2.5). Definition 2.2. By a mild solution of the identification problem (IP) we mean a pair (z, u) X × C([-, 1]; X) satisfying u(t) = h(t) for t [ -, 0 ], u(t) = h(0) + S(t - s) [(s, us )z + f (s, us )] for t [ 0, 1 ], and the additional condition (1.2). Remark 2.1. Clearly, each strict solution of (IP) is a mild solution too, buhe converse statement is no longer true. Moreover, each mild solution of (IP) is a generalized solution too. The converse of the last statement hol also true under the additional condition that A . We notice thahis condition is satisfied, whenever A generates a C0 -semigroup having exponential decay. 7 Indeed, we have: IDENTIFICATION OF A SOURCE Lemma 2.2. Let us assume that A , D(A), hC([-, 0]; X) satisfies h(0) D(A) and (2.4) hol true. If u is a generalized solution of the identification problem (IP), then u is a mild solution of the problem (1.1), with z = F (u) given by (2.5), and, in addition, u satisfies (1.2). Also the proof of Lemma 2.2 is very similar to the proof of the corresponding Lemma 2.2 in Lorenzi and Vrabie [9] and therefore we omit it. 2.3. The main assumptions The hypotheses we need are listed below. (HA ) A : D(A) X X generates a C0 -semigroup {S(t); t 0} and (A1 ) the resolvent of A, i.e. (I - A) is compact ; (A2 ) there exists > 0 such that S(t) e-t for each t 0 ; (H ) : [ 0, 1 ] × C([ -, 0 ]; X) R is a functional satisfying (1 ) there exists > 0 such that |(t, v) - (t, v) |t - t| + v - v (t, v), (t, v) [ 0, 1 ] × C([ -, 0 ]; X) ; (2 ) is bounded from below by a strictly positive number and it is bounded from above, i.e., m1 := inf{(t, v); (t, v) [ 0, 1 ] × C([ -, 0 ]; X)} > 0 m2 := {(t, v); (t, v) [ 0, 1 ] × C([ -, 0 ]; X)} < + ; (Hf ) f : [ 0, 1 ] × C([ -, 0 ]; X) X is a C 1 -function and there exiswo constants i > 0, i = 1, 2, such that (f1 ) f (t, v)-f (t, v) 1 |t - t| + v - v (t, v) [ 0, 1 ] × C([ -, 0 ]; X) ; (f2 ) f (t, 0) 2 for each t [ 0, 1 ] ; C([ -,0 ];X) C([ -,0 ];X) for each (t, v), ALFREDO LORENZI and IOAN I. VRABIE (H ) W 1,1 ([ 0, 1 ]; R) \ {0} , (t) 0 for each t [ 0, 1 ]. Remark 2.2. (i) If A satisfies (A2 ) then A + I is dissipative see Pazy [11, Theorem 4.3] and thus A . As a consequence, A is injective. (ii) If (2 ) in (H ) and (H ) hold, then there exists a constant m > 0 such that m for each u C([ -, 1 ]; X). Indeed, if u C([ -, 1 ]; X), then m1 (t)dt := m > 0. So J, defined by (2.6), satisfies |J(u)| for each u C([ -, 1 ]; X). 2.4. Preliminary estimates Proposition 2.1. Let (H ), (Hf ) and (H ) be satisfied. Then the operator F , defined by (2.5), with J being given by (2.6), is Lipschitz continuous on bounded subsets in (2.8) C[h] ([ -, 1 ]; X) = {u C([ -, 1 ]; X); u(t) = h(t) for t [ -, 0 ]} and there exiswo constants L0 > 0 and M0 > 0 such that (2.9) F (u) L0 u C([-,1 ];X) 1 m + M0 for each u C[h] ([ -, 1 ]; X). Proof. Let us observe firshat F is decomposed as F (u) = J(u)P (u) for each u C[h] ([ -, 0 ]; X), where J : C([ -, 1 ]; X) R is defined by (2.6) and P : C[h] ([ -, 1 ]; X X is given by P (u) := + (1)u(1) - (t)f (t, ut ) dt IDENTIFICATION OF A SOURCE for each u C[h] ([ -, 0 ]; X), being defined in (2.6), i.e. = -A - (0)h(0). We begin by showing thahe mapping u J(u), defined by by (2.6), is Lipschitz with constant (ii) in Remark 2.2, we deduce 1 . Indeed, from (H ) and (H ) and |J(u) - J(u)| - = u-u for each u, u C[h] ([ -, 0 ]; X). Next, since for each u, u C[h] ([ -, 0 ]; X) and t [ -, 0 ], u(t) = u(t) = h(t), we have (2.10) Therefore (2.11) |J(u) - J(u)| u-u = u-u . u-u for each u, u C[h] ([ -, 0 ]; X). Ahis point, let us observe that P can be rewritten as P (u) = +(1)u(1)- (t)u(t) dt- (t)[f (t, 0)+f (t, ut )-f (t, 0)] dt, where is given by (2.6). So, we deduce P (u) + (1) + (2.12) + C([ -,0 ];X) dt f (·, 0) L1 (0,1;R) (t) ut u (1) + + ALFREDO LORENZI and IOAN I. VRABIE for each u C[h] ([ -, 1 ]; X). We also have P (u) - P (u) (1) u(1) - u(1) + | (t)| u(t) - u(t) dt L1 (0,1;R) (t) f (t, ut ) - f (t, ut ) dt (1) + u-u (t)[ ut - ut C([ -,0 ];X) dt (1) + L1 (0,1;R) L1 (0,1;R) u-u for each u, u C[h] ([ -, 1 ]; X) and so (2.13) P (u) - P (u) (1) + L1 (0,1;R) L1 (0,1;R) u-u . On the other hand, we have F (u) - F (u) J(u)P (u) - J(u)P (u) + J(u)P (u) - J(u)P (u) . Since u C([ -,0 ];X) + u C([ 0,1 ];X) , from (2.10), (2.11), (2.12) and (2.13), we get F (u) - F (u) + P (u) u - u + 1 P (u) - P (u) m k0 + k1 u L1 (0,1;R) u-u , 1 (1) + m L1 (0,1;R) u-u where (we recall that, by (2.6), = A + (0)h(0) ) (2.14) k0 := A + (0)h(0) h C([ -,0] ;X) k1 := (1) + L1 (0,1;R) L1 (0,1;R) . L1 (0,1;R) , So, if the set K C[h] ([ -, 0 ]; X) is bounded, say by r > 0, we conclude that F (u) - F (u) L u - u , IDENTIFICATION OF A SOURCE for each u, u K, where L := (k0 + k1 r) + 1 (1) + m L1 (0,1;R) L1 (0,1;R) Finally, from Remark 2.2 and (2.12), we deduce that F (u) L0 u for each u C[h] ([ -, 1 ]; X), with (2.15) L0 := k1 , m M0 := k2 , m + M0 where k1 is given by (2.14) and k2 := A + (0)h(0) L1 (0,1;R) . Thus F satisfies the linear growth condition (2.9) and this completes the proof. 3. The main resulheorem 3.1. Lehe assumptions (HA ), (H ), (Hf ) and (H ) be satisfied. Let us assume that (3.1) m2 L0 < , where L0 > 0 is given by (2.15). Let h C([ -, 0 ]; X) with h(0) D(A). Then the identification problem (IP) has at least one mild solution (z, u), where u is a Lipschitz continuous mild solution of the direct problem (3.2) and (3.3) z = F (u), u (t) = Au(t) + (t, ut )F (u) + f (t, ut ), u(t) = h(t), t [ 0, 1 ], t [ -, 0 ], F being defined by (2.5) and (2.6). ALFREDO LORENZI and IOAN I. VRABIE Remark 3.1. Condition (3.1) says that is large if compared with the Lipschitz constant of f , the L1 -norms of and and the value of at 1. See the definition of L0 , i.e., (2.15) and (2.14). For instance, if 1, i.e., if (t) dt is the Lebesgue measure, then, according to Remark 2.2, (3.1) is satisfied if m2 (1 ) < , m1 where > 0, m1 > 0, m2 > 0 and 1 > 0 are defined in the hypotheses (HA ), (H ) and respectively (Hf ). If (3.1) is satisfied then, for each h C([ -, 0 ]; X), there exists rh > 0 such that (3.4) 4 h C([ -,0 ];X) m2 (L0 r + M0 ) r - 1 for each r rh , where L0 > 0, M0 > 0 are given by (2.15) and 2 > 0 by (Hf ). 3.1. The idea of the proof It is easy to observe that (IP) has a mild (strict) solution if and only if (3.2) has a mild (strict) solution u and z is given by (3.3). Let h C([ -, 0 ]; X) with h(0) D(A) and D(A) be arbitrary but fixed, let r > max{0, h C([ -,0 ];X) } and, for any L > 0, let us define the sets CL ([ -, 1 ]; X) := {u C([ -, 1 ]; X); u(t)-u(s) L|t-s|, t, s [ 0, 1 ]} and (3.5) K := {u C[h] ([ -, 1 ]; X) CL ([ -, 1 ]; X); u r}, C[h] ([ -, 1 ]; X) being defined by (2.8). Clearly K is nonempty, closed and convex in C([ -, 1 ]; X). Let us define the operator S : K C([ -, 1 ]; X) by (3.6) S(x) := u, where u C([ -, 1 ]; X) is the unique mild solution of the semilinear delay evolution problem (3.7) u (t) = Au(t) + (t, ut )F (x) + f (t, ut ), t [ 0, 1 ] u(t) = h(t), t [ -, 0 ]. IDENTIFICATION OF A SOURCE We will show that, under the hypotheses of Theorem 3.1, we can choose r > 0 and L > 0 such thahe operator S maps K into itself, is continuous and S(K) is relatively compact in C([ -, 1 ]; X). Then, by Schauder's Theorem, S has at least one fixed point u K, which provides a mild solution (z, u) for (IP), where z = F (u). 4. Proof of the main result The Bellman-type inequality below is a variant of Lemma 4.3 in Burlica and Rosu [2]. Since its proof is almost identical with the one of Lemma 4.3 ¸ in Burlica and Rosu [2], we do not give details. ¸ Lemma 4.1. Le > 0 and let y : [ -, T ] R+ and 0 , : [ 0, T ] R+ be continuous functions, 0 being nondecreasing. If (4.1) y(t) 0 (t) + (s) ys C([ -,0 ]:R) for each t [ 0, T ], then (4.2) y(t) (t) + (s)(s)exp () d , for each t [ 0, T ], where (t) := y0 for each t [ 0, T ]. We will show how to choose L > 0 such thahe operator S, defined as above, maps K given by (3.5) into itself. More precisely, we will prove Lemma 4.2. Let (HA ), (H ), (Hf ) and (H ) be satisfied. Let h C 1 ([ -, 0 ]; X) with h(0) D(A) and let us assume that (3.1) hold. Then, there exist r > 0 and L > 0 such thahe operator S, defined by (3.6), maps the set K defined by (3.5) into itself, is continuous and S(K) is compact. Proof. Let u be the unique mild solution of the problem (3.7). Since, by (A2 ) in (HA ), A generates a contraction semigroup {S(t); t 0} satisfying S(t) e-t for each t 0, using the variation of constants formula, (2 ) in H and (Hf ), we deduce u(t) e-t u(0) + + 0 (t) e-(t-s) m2 F (x) us . ALFREDO LORENZI and IOAN I. VRABIE Whence et u(t) u(0) + es m2 F (x) us . Let us fix an r > h C([ -,0 ];X) which will be chosen more precisely later on. By virtue of (2.9), F (x) L0 x + M0 L0 r + M0 , for each x C[h] ([ -, 1 ]; X) with x r, where L0 and M0 are given by (2.15). Denoting by k = m2 (L0 r + M0 , the preceding inequality can be rewritten as et u(t) u(0) + k t e + t 0 1 es us for each t [ 0, 1 ]. Buhis inequality is of the choice of y, 0 , and as specified below: y(t) = et u(t) , k (s) = u(0) + (es - 1) , 0 (s) = 1 , Observing that us y0 C([ -,0 ];X) form (4.1) for the specific t [ -, 1 ] s [ 0, 1 ] s [ 0, 1 ]. = ( u(·) ) s = (e C(([ -,0 ];R) for each s [ 0, 1 ], h C([ -,0 ];X) , u(·) )0 from Lemma 4.1, we deduce thahe function y, defined as before, satisfies (4.2). Equivalently, t u(t) satisfies u(t) e-t (t) + e-t t (s)(s)exp () d for each t [ 0, 1 ]. Recalling the definition of , see Lemma 4.1 and taking into accounhat u(0) = h(0) h C([ -,0 ];X) we conclude that u(t) e-t 2 h (4.3) + e-t k t e k 2 h C([ -,0 ];X) + (es - 1) 1 e1 (t-s) C([ -,0 ];X) IDENTIFICATION OF A SOURCE for each t [ 0, 1 ]. Since, by (3.1), - 1 > 0, the second term on the right-hand side of the preceding inequality can be estimated as e-t 2 h C([ -,0 ];X) t 0 1 e-( )t = 21 e-( )t k s (e - 1) 1 e1 (t-s) k 2 h C([ -,0 ];X) + es e s + k 1 - e-( )t 1 - e t h C([ -,0 ];X) 1 ( - 1 ) 1 k . 2 h + ( - 1 ) From (4.3) and the last relation, taking into accounhat, for t 0, e-( )t 1 - e t 1, 1 - e-( )t 1, we get u(t) 4 h k - 1 for each t [ 0, 1 ], Now, recalling that k = m2 (L0 r + M0 ) , we conclude that (4.4) u(t) 4 h m2 (L0 r + M0 ) - 1 for each t [ -, 1 ]. From (3.1) and the last part of Remark 3.1, it follows thahere exists rh > 0 such that (3.4) hol for each r rh . Let us fix r max{rh , h C([ -,0 ];X) }. From (4.4), we deduce that for each x C[h] ([ -, 1 ]; X), satisfying (4.5) we have (4.6) u C([ ,1 ];X) C[h] ([ -,1 ];X) r, C[h] ([-,1];X) which shows that S maps the set {x C[h] ([-, 1]; X); x into itself. r} ALFREDO LORENZI and IOAN I. VRABIE Next, we show how to choose the Lipschitz constant L > 0 such thahe set K, defined by means of (3.5), is invariant under S, S(K) is relatively compact in C[h] ([ -, 1 ]; X) and S is continuous from K into K. To this aim, let us observe that, for each x C[h] ([ -, 1 ]; X), we have (4.7) u(t) = S(t)u(0) + (s, us )S(t - s)F (x) + S(t - s)f (s, us) , and u(t + ) = S(t)u() + (s, us )S(t + - s)F (x) S(t + - s)f (s, us ) = S(t)u() + (s + , us+ )S(t - s)F (x) S(t - s)f (s + , us+ ) . So, we get u(t + ) - u(t) e-t u() - u(0) e-(t-s) + us+ - us C([ -,0 ];X) F (x) F (x) e-(t-s) + us+ - us C([ -,0 ];X) for each (0, 1) and t [ 0, 1 - ]. Whence u(t + ) - u(t) e-t u() - u(0) + 1 - e-t C([ -,0 ];X) . F (x) + [ F (x) ] e-(t-s) us+ - us Thanks to (2.9), we have F (x) L0 x + M0 L0 r + M0 , r, where L0 and M0 are for each x C[h] ([ -, 1 ]; X) with x given by (2.15). So, setting (4.8) := (L0 r + M0 ) , IDENTIFICATION OF A SOURCE the above inequality lea to et u(t + ) - u(t) u() - u(0) + t e C([ -,0 ];X) (4.9) es us+ - us Thanks to Lemma 4.1, the function y, defined above, satisfies (4.2) on [ 0, 1 - ] with given by (s) = u() - u(0) + (es - 1) + u - u0 C([ -,0 ];X) . Equivalently, the function t u(t + ) - u(t) satisfies u(t + ) - u(t) e-t (t) + e-t for each (0, 1) and t [ 0, 1 - ]. Clearly, (4.9) y(t) = et u(t + ) - u(t) , s 0 (s) = u() - u(0) + (e - 1) , (s) = , is of the form (4.1), with t [ -, 1 - ], s [ 0, 1 - ], s [ 0, 1 - ]. (s)e(t-s) for t [ 0, 1 - ]. Obviously (s) 2 u - u0 and therefore t e t 2 u - u0 C([ -,0 ];X) + es e(t-s) . + e-t 0 Estimating the second term on the right-hand side, we get u(t + ) - u(t) e-t 2 u - u0 C([ -,0 ];X) C([ -,0 ];X) s e e-t 2 u - u0 t 0 = e-(-)t = e-(-)t s e e(t-s) 2 u - u0 C([ -,0 ];X) + es e-s C([ -,0 ];X) 1 - e-t 2 u - u0 C([ -,0 ];X) C([ -,0 ];X) + e(-)t - 1 ( - ) 2e-(-)t u - u0 2 1 - e-(-)t . ( - ) ALFREDO LORENZI and IOAN I. VRABIE Let (4.10) Then, we have both e-(-)t µ, and 1 - e-(-)t µ. - µ = {e-(-)t ; t [ 0, 1 ]}. To prove the last inequality, first, let us observe that we may assume with no loss of generality that - = 0. Indeed, if either > or < , we have nothing to do. If = , then we can slightly diminish > 0 such that both (3.1) and - = 0 hold true. Next, by Lagrange MeanValue Theorem applied to the function x e-x either on [ 0, -( - )t ] if - < 0 or on [ -( - )t, 0 ] if - > 0, it follows thahere exists t1 [ 0, 1 ] such that, in both cases mentioned before, we have 1 - e-(-)t e-(-)t1 = ( - )t = e-(-)t1 t e-(-)t1 µ. - - Accordingly, we get e-t 2 u - u0 C([ -,0 ];X) s e e(t-s) 2µ u - u0 Finally, we obtain C([ -,0 ];X) 2 µ . u(t + ) - u(t) 2 u - u0 2 µ , µ u - u0 C([ -,0 ];X) + C([ -,0 ];X) and thus (4.11) u(t + ) - u(t) 2(1 + µ) u - u0 C([ -,0 ];X) (1 + µ) , for each (0, 1) and t [ 0, 1 - ]. We estimate now u - u0 C([ -,0 ];X) t[ -,0 ] u(t + ) - u(t) . IDENTIFICATION OF A SOURCE Le [ -, 0 ] and (0, 1). We have two complementary cases. Case 1. - t + 0. Since h C 1 ([-, 0]; X), we have (4.12) u(t + ) - u(t) h(· + ) - h(·) C([-,0];X) C([ -,0 ];X) . Case 2. - t 0 < t + . Then we get (4.13) u()-u(t) = u()-h(t) u()-u(0) + h(0)-h(t) . Ahis point, since 0 < t + < , from (4.7) and (2.9), we deduce u(t + ) - u(0) S(t + )u(0) - u(0) [(s, us ) F (x) + f (s, us ) ] Ah(0) + m2 (L0 r + M0 ) r (t + ) Ah(0) + m2 (L0 r + M0 ) r . (4.14) Further, since - < t 0, we have |t| < and thus h(0) - h(t) h C([ -,0],];X) |t| C([ -,0],];X) . Summing up, from (4.12), (4.13) and (4.14), we get u(t + ) - u(t) Ah(0) + m2 (L0 r + M0 ) r + h C([-,0];X) for each (0, 1) and t [ 0, 1 - ]. So, we have proved the estimate (4.15) u(· + ) - u C([-,0];X for each x C[h] ([ -, 1 ]; X) with x r, each t [ 0, 1 ] and for each (0, 1) and t [ 0, 1 - ], where L is given by L=2(1 + µ) Ah(0) +m2 (L0 r + M0 )+1 r + h (1 + µ) (4.16) , + C([-,0];X) being defined by (4.8) and µ by (4.10). Let us define K as in (3.5), i.e., K := x C[h] ([ -, 1 ]; X) CL ([ -, 1 ]; X); where r > 0 and L > 0 are chosen as above and CL ([-, 1]; X) := {u C([-, 1]; X); u(t)-u(s) L|t-s|, t, s [0, 1]}. u ALFREDO LORENZI and IOAN I. VRABIE From (4.6) and (4.15), we conclude that S maps K into itself. To prove the relative compactness of S(K) in C([ -, 1 ]; X), let us note that, thanks to the fachat u(t) = h(t) for each u K and each t [ -, 0 ], it suffices to show that K is relatively compact in C([ 0, 1 ]; X). Clearly S(K) is equicontinuous from the right on [ 0, 1), containing only Lipschitz functions with the very same constant L. Furthermore, the set {t (t, ut )F (x) + f (t, ut ); x K, u = S(x)} is uniformly bounded on [ 0, 1 ] and so Lebesgue uniformly integrable in L1 (0, 1; X). Thus, S(K) satisfies all the hypotheses of "(c1 ) implies (c2 )" part in Theorem 2.1. Whence S(K) is relatively compact in C([ 0, 1 ]; X). Finally, let (xp )pN is a sequence in K which converges uniformly to x K. As F is Lipschitz continuous, it follows that lim F (xp ) = F (x). Now, if some subsequence of (S(xp ))pN converges to a certain function u, recalling that , F and f are jointly continuous, it follows that u must coincide with S(x). But {S(xp ); p N} is relatively compact in C([ 0, 1 ]; X). Using this remark and a usual argument by contradiction, we deduce thahe sequence itself converges to S(x). So S is continuous and this completes the proof of Lemma 4.2. Now, we can pass to the proof of Theorem 3.1. Proof. By Lemma 4.2 and Schauder's Fixed Poinheorem, we conclude thahe operator S, defined by (3.6), has at least one fixed point u K. But each fixed point of S is a mild solution of the problem (3.2) which, by the definition of K, is Lipschitz continuous with constant L. Obviously u is a generalized solution of (IP). Since A satisfies (A2 ), by virtue of (i) in Remark 2.2 combined with Lemma 2.2, it follows that (z, u), with z given by (3.3), is a mild solution of (IP) and this completes the proof of Theorem 3.1. 5. Uniqueness and continuous dependence The aim of this Section is to state and prove a uniqueness and continuous dependence result referring to mild solutions of (IP) in the sense of Definition 2.2. First, some preliminary evaluations are needed. As u satisfies IDENTIFICATION OF A SOURCE (2.7), where F depen on u but also on and h, we can express z in terms of u, , h as: z = F (u, , h), where F (u, , h) is defined (cf. (2.4), (2.5) and (2.6)) by F (u, , h)=J(u, h) (, h)+(1)u(1)- (t)f (t, ut ) dt , (, h) = -A - (0)h(0), J(u, h) = Recall now that, by (ii) in Remark 2.2, we have (5.1) J(u, h) m , for all u C([ 0, 1 ]; X) and h C([-, 0]; X). Let (, h), (, h) D(A)×C([ -, 0 ]; X) be such that h(0), h(0) D(A) and let u and u be two mild solutions of (IP), the first one corresponding to (, h) and the second one to (, h). We emphasize that, for the time being, we do not know that u and u are uniquely determined. Let us observe that, in order to prove both the uniqueness and the continuous dependence of u on (, h), we need some sharp evaluations of the difference u - u. To obtain this, we have firso evaluate |J(u, h)-J(u, h)| and F (u, , h)-F (u, , h) . 5.1. An estimate of F (u, , h) - F (u, , h) First, we observe that, for each h, h C([ -, 0 ]; X), u C[h] ([ -, 1 ]; X), u C[h] ([ -, 1 ]; X) and each t [ 0, 1 ], we have ut - ut C([ -,0 ];X) s[ -,0 ] u(t + s) - u(t + s) u(t + s) - u(t + s) C([ 0,1 ];X) , t[ 0,1 ] s[ -,0 ] = max{ u - u u-u C([ -,0 ];X) } C([ 0,1 ];X) + C([ -,0 ];X) . ALFREDO LORENZI and IOAN I. VRABIE Whence, from (5.1), (1 ) in (H ) and (f1 ) in (Hf ) (cf. Subsection 2.3), we deduce |J(u, h) - J(u, h)| (t)|(t, ut ) - (t, ut )| dt C([ -,0 ];X) dt C([ 0,1 ];X) (t) ut - ut L1 (0,1;R) u-u + C([-,0];X) dt for all h, h C([ -, 0 ]; X), u C[h] ([ -, 1 ]; X) and u C[h] ([ -, 1 ]; X). Likewise (t) f (t, ut ) - f (t, ut ) dt L1 (0,1;R) u-u C([ 0,1 ];X) + C([-,0];X) dt for all h, h C([ -, 0 ]; X), u C[h] ([ -, 1 ]; X) and u C[h] ([ -, 1 ]; X). Then, we get F (u, , h) - F (u, , h) |J(u, h) - J(u, h)| (, h) +(1) u(1) | (t)| u(t) dt + (t) f (t, ut ) dt + |J(u, h)| ( - , h - h) + (1) u(1) - u(1) (5.2) | (t)| u(t) - u(t) dt + 0 L1 (0,1;R) 1 (t) f (t, ut ) - f (t, ut ) dt + 1 C([-,0];X) u-u C([ 0,1 ];X) (, h) + (1) u(1) + | (t)| u(t) dt + (t) f (t, ut ) dt L1 (0,1;R) ] + m We note that ( - , h - h) + [(1) + L1 (0,1;R) u-u C([ 0,1 ];X) u-u C([ 0,1 ];X) + C([-,0];X) ( - , h - h) - D(A) C([ 0,1 ];R) C([ -,0 ];X) , for each (, h) D(A) × C([ -, 0 ]; X). IDENTIFICATION OF A SOURCE 5.2. The increments of the forcing term in (3.2) Let us denote by G the forcing term in (3.2), i.e., G(u, , h)(t) = (t, ut )F (u, , h) + f (t, ut ), for t [ 0, 1 ]. Let us recall that, in view of (2.14) and (2.9), we have (5.3) F (u, , h) L0 u C([ 0,1 ];X) + M0 (, h), where L0 and M0 (, h) are given by (2.15). We notice that L0 is independent of both and h. Moreover, the mapping (, h) M0 (, h), defined as in (2.15), is continuous from D(A) × C([ -, 0 ]; X), D(A) being endowed with the graph norm and C([ -, 0 ]; X) with the usual -norm. From the identity G(u, , h)(t) = (t, ut )F (u, , h) + f (t, ut ) - f (t, 0) + f (t, 0), for all t [ 0, 1 ], combined with (5.3), we deduce the following estimate (recall that, by (f2 ) in (Hf ), we have f (·, 0) C([ 0,1 ];X) 2 ): G(u, , h)(t) m2 L0 u (5.4) +1 h C([-,0];X) C([ 0,1 ];X) + M0 (, h) u C([ 0,1 ];X) C([ 0,1 ];X) (m2 L0 ) u + m2 M0 (, h) h C([-,0];X) . 5.3. An a priori estimate for the mild solution of the problem (IP) It is immediate to check ouhat (IP) is equivaleno the following integral equation u(t) = S(t)h(0) + S(t - s)G(u, , h)(s) (5.5) =: S(t)h(0) + N (u, , h)(t), for t [ 0, 1 ]. Let us assume for the momenhat: (5.6) where S(·) L1 < (m2 L0 ) , (5.7) S(·) L1 := . ALFREDO LORENZI and IOAN I. VRABIE For a sufficient condition in order that (5.6) be satisfied see Remark 5.1 below. Then, from (5.4), we deduce u(t) S(t)h(0) + (m2 L0 ) u + m2 M0 (, h) h C([ 0,1 ];X) t C([-,0];X) , for each t [ 0, 1 ]. Whence, we get u C([ 0,1 ];X) S(·)h(0) C([0,1];X) +(m2 L0 +1 ) C([0,1];X) S(·) L1 (5.8) + m2 M0 (, h) h This, in turn, implies u (5.9) where C([ 0,1 ];X) C([-,0];X) S(·) m h(0) S(·) 3 S(·) L1 C([ 0,1 ];) + m m2 M0 (, h) 3 h C([-,0];X) =: r1 (, h), m3 := 1 - (m2 L0 ) S(·) L1 and S(·) L1 is defined in (5.7). Clearly, by (5.6), we deduce that m3 > 0. In particular, we have shown that N (u, , h) (5.10) C([ 0,1 ];X) (m2 L0 ) u C([-,0];X) C([ 0,1 ];X) S(·) L1 + m2 M0 (, h) h S(·) Recalling the expression of m3 , we conclude that r1 (, h) is given by r1 (, h) = + h(0) S(·) C([ 0,1 ];) 1 - (m2 L0 ) S(·) L1 m2 M0 (, h) h C([-,0];X) L1 S(·) L1 1 - (m2 L0 ) S(·) Consequently, from (5.3), (5.8) and (5.10), we get (5.11) F (u, , h) C([ 0,1 ];X) L0 r1 (, h) + M0 (, h) =: r2 (, h). Summing up, we have proved: IDENTIFICATION OF A SOURCE Lemma 5.1. Under the condition (5.6) each solution of the equation (5.5) satisfies the estimate (5.9). Remark 5.1. Assume that (5.12) S(t) e-t , for each t > 0 and some > 0 satisfying m2 L0 . Then (5.13) S(·) L1 1 - e-t < (m2 L0 ) and thus (5.6) hol true. Lemma 5.2. Under the condition (5.12) each solution of the equation (5.5) satisfies the estimate (5.9). Ahis point, let us observe that, u and u are the mild solutions of the following problems: u (t) = Au(t) + G(u, , h)(t), u(0) = h(0), and u (t) = Au(t) + G(u, , h)(t), u(0) = h(0). t (0, 1), t (0, 1), 5.4. A new estimate of F (u, , h) - F (u, , h) From (5.2) and (5.9), we have F (u, , h) - F (u, , h) L1 (0,1;R) u-u C([0,1];X) + C([-,0];X) (, h) (5.14) + (1) + + m L1 (0,1;R) r1 (, h) + L1 (0,1;R) 2 L1 (0,1;R) ( - , h - h) + (1) + L1 (0,1;R) u-u C([ 0,1 ];X) u-u C([ 0,1 ];X) + C([-,0];X) C([-,0];X) =:r3 (, h) u - u C([ 0,1 ];X) +r4 (, h) A - A + ALFREDO LORENZI and IOAN I. VRABIE C([ 0,1 ];X) where (recall that, by (f2 ) in (Hf ), f (·, 0, 0) r3 (, h) = L1 (0,1;R) 2 ) A + (0) h C([-,0];X) L1 (0,1;R) + (1) + (5.15) r4 (, h) = max L1 (0,1;R) r1 (, h) A + (0) h r1 (, h) + L1 (0,1;R) + m (1) + L1 (0,1;R) L1 (0,1;R) C([-,0];X) + (1) + L1 (0,1;R) 2 + m max{1, (0)} 5.5. An estimate of G(u, , h) - G(u, , h) Starting from the the identity G(u, , h)(t) - G(u, , h)(t) = (t, ut ) - (t, ut ) F (u, , h) + (t, ut ) F (u, , h) - F (u, , h) + f (t, ut ) - f (t, ut ), for each t [ 0, 1 ], from (2 ) in (H ), (5.11), (5.14) and (5.15), we deduce the following estimate: G(u, , h) - G(u, , h) C([ 0,1 ];X) C([ 0,1 ];X) C([-,0];X) r2 (, h) + m2 r3 (, h) u - u + m2 r4 (, h) ] A - A + h - h Let us observe that, for all t [ 0, 1 ], N (u, , h)(t) - N (u, , h)(t) = S(t - s) G(u, , h)(s) - G(u, , h)(s) . So, for all t [ 0, 1 ], we have N (u, , h)(t) - N (u, , h)(t) (5.16) r2 (, h) + m2 r3 (, h) u - u C([ 0,1 ];X) S(·) L1 + m2 r4 (, h) ] A - A + h - h C([-,0];X) S(·) IDENTIFICATION OF A SOURCE 5.6. Uniqueness and continuous dependence on the data Let us denote by C0,A ([ -, 0 ]; X) the linear space of all functions h C([ -, 0 ]; X) satisfying h(0) D(A). Endowed with the norm h 0,A = { h(t) ; t [ -, 0 ]} + h(0) + Ah(0) } this is a Banach space. As r1 , . . . r4 , M0 : D(A) × C0,A ([ -, 0 ]; X) (0, +) are continuous whenever D(A) and C0,A ([ -, 0 ]; X) are endowed, respectively, with the graph norm and the norm · 0,A , from the estimates above, we deduce: Theorem 5.1. In addition to the hypotheses of Theorem 3.1 assume that (5.17) and (5.18) > max {(m2 L0 ), r2 (, h) + m2 r3 (, h) } S(·) L1 < min {(m2 L0 ) , r2 (, h) + m2 r3 (, h) are satisfied, where r2 (, h) is given by (5.11) and r3 (, h) is given by (5.15). Then, the mild solution (z, u), whose existence is ensured by Theorem 3.1, is unique and the application (, h) (z, u), from D(A) × C0,A ([ -, 0 ]; X) to X × C([ 0, 1 ]; X), is locally Lipschitz. Here D(A) is endowed with the graph norm, C0,A ([ -, 0 ]; X) with the norm · 0,A defined as above and X × C([ 0, 1 ]; X) with the norm · + · C([ 0,1 ];X) . Proof. If (5.17) and (5.18) are satisfied, then (5.13) hol true. So, all the estimates obtained before hold true. Choose now (, h) D(A) × C0,A ([ -, 0 ]; X) satisfying (5.17) and (5.18). For a fixed satisfying (3.1) this is possible if satisfies (5.18) with (, h) = (0, 0). Indeed, r (0, 0) = 2 m2 L1 (0,1;R) S(·) L1 , 1 1 - (m2 L0 ) S(·) L1 2 r2 (0, 0) = L0 r1 (0, 0) + L1 (0,1;R) , m -2 r3 (0, 0) = 2 m L1 (0,1;R) |(1)| + L1 (0,1;R) r1 (0, 0) +2 L1 (0,1;R) + m |(1)| + L1 (0,1;R) L1 (0,1;R) . So, if and h are small, satisfies (5.18). ALFREDO LORENZI and IOAN I. VRABIE Then u N (u, , h) is a contraction in the metric space K(, h) = {u C([ 0, 1 ]; X); u C([ 0,1 ];X) r1 (, h)}. Therefore, for any pair satisfying (5.17) and (5.18), the equation (5.5) admits a unique solution u = U (, h) C([ 0, 1 ]; X). Moreover, the operator U depen continuously on the set of consisting of all pairs (, h) satisfying (5.17) and (5.18). Since r1 r4 depend continuously on (, h) D(A) × C0,A ([ -, 0 ]; X) if D(A) is endowed with the graph norm and on C0,A ([ -, 0 ]; X) with the norm · 0,A , it follows thahe set D of all pairs satisfying (5.17) and (5.18) is open. So, if (, h), (, h) D, from (5.5) and (5.16), we deduce that U (, h) - U (, h) C([ 0,1 ];X) C([ 0,1 ];X) r2 (, h) + m2 r3 (, h) U (, h)(t) - U (, h) C([-,0];X) + m2 r4 (, h) ] A - A + h - h implying that U (, h) - U (, h) + where C([ 0,1 ];X) S(·) L1 , 1 - r5 (, h) m2 r4 (, h) ] A - A C([-,0];X) S(·) L1 , r5 (, h) = r2 (, h) + m2 r3 (, h) S(·) Whence U is locally Lipschitz and this completes the proof. 6. An application to an integrodifferential parabolic problem Let in Rn be a bounded domain whose boundary is of class C 2 . We will now apply the abstract results of the Sections 3 and 6 to recover the unknown function u : [0, T ] × R and the coefficient z : R in the following integrodifferential parabolic identification problem: (IP1 ) Find z and u : [ -, 1 ] satisfying u (t, x) = Au(t, x) + z(x)(t, ut ) t +f (t, ut )(x), (t, x) [ 0, 1 ] × , (6.1) Bu(t, x) = 0, (t, x) [ 0, 1 ] × , u(t, x) = h(t, x), (t, x) (-, 0] × , IDENTIFICATION OF A SOURCE and the additional condition (6.2) (t)u(t, x) dt = (x), x , where the functional and the operator f are as in the two cases described below. Case 1. The functional is of the form (t, v) = t, (-,0)× v(s, y) , for each t [ 0, 1 ] and each v C([ -, 0 ]; ), and f (t, v)(x) = g, x, v(s, x) + g1 t, x, y, v(s, y) , = f0 (t, v)(x) + f1 (t, v)(x), for each t [ 0, 1 ], each v C([ -, 0 ]; ) and for a.e. x , where : [ 0, 1 ] × R R, g0 : [ 0, 1 ] × × R R and g1 : [ 0, 1 ] × × × R R are given functions satisfying the properties: (H ) C 0,1 ([ 0, 1 ] × R)2 and there exiswo positive constants m1 , m2 such that m1 (t, v) m2 for all (t, v) [ 0, 1 ] × R; (H g0 ) for each (t, v) [ 0, 1 ] × R, the function x g0 (t, x, v) is measurable and there exists l0 such that ~ ~ ~ |g0 (t, x, v) - g0 (t, x, v )| l0 |t - t| + |v - v | ~ ~ for all t, t [ 0, 1 ], v, v and a.e. for x ; ~ (H g1 ) for each (t, v) [ 0, 1 ] × R, the function (x, y) g1 (t, x, y, v) is measurable and there exists l1 : × R such that, a.e. for y , x l1 (x, y) belongs to , y belongs to and ~ ~ |g1 (t, x, y, v) - g1 (t, x, y, v )| l1 (x, y) |t - t| + |v - v | ~ ~ ~ for all t, t [ 0, 1 ], v, v and for a.e. x and y . ~ C 0,1 ([ 0, 1 ] × R) denotes the Banach space of all Lipschitz-continuous functions in [ 0, 1 ] × R. ALFREDO LORENZI and IOAN I. VRABIE Case 2. The functional is of the form (t, v) = (-,0)× (t, s, y, v(s, y)) , for each t [ 0, 1 ] and each v C([ -, 0 ]; ), and f (t, v)(x) = g0 (t, s, x, v(s, x)) + (-,0)× g1 (t, s, x, y, v(s, y)) , = f0 (t, v)(x) + f1 (t, v)(x), for each t [ 0, 1 ], each v C([ -, 0 ]; ) and a.e. for x , where : [ 0, 1 ] × [ -, 0 ] × × R R, g0 : [ 0, 1 ] × [ -, 0 ] × × R R and g1 : [ 0, 1 ] × [ -, 0 ] × × × R R are given functions satisfying the properties: (H ,1 ) for all (t, v) [ 0, 1 ] × R, the function (s, y) (t, s, y, v) is measurable; (H ,2 ) there exist 1 , 2 L1 ((-, 0) × ) \ {0} such that 0 1 (s, y) (t, s, y, v) 2 (s, y) for all (t, v) [ 0, 1 ] × R and for a.e. (s, y) (-, 0) × ; (H ,3 ) there exists a nonnegative function l Lq (( 0, 1 ) × ) such that ~ ~ |(t, s, y, v) - (t, s, y, v )| l(s, y) |t - t| + |v - v | ~ ~ ~ for all t, t [ 0, 1 ], v, v and for a.e. (s, y) ( 0, 1 ) × ; ~ (H g0 ,1 ) for all (t, v) [ 0, 1 ] × R, the function (s, y) g0 (t, s, x, v) is measurable; (H g0 ,2 ) there exists a function l0 L1 ( 0, 1 ) such that ~ ~ |g0 (t, s, x, v) - g0 (t, s, x, v )| l0 (s) |t - t| + |v - v | ~ ~ ~ for all t, t [ 0, 1 ], v, v and for a.e. (s, x) (0, 1) × ; ~ (H g0 ,3 ) the function t 0 - g0 (t, s, ·, 0) is bounded on (0, 1) × ; IDENTIFICATION OF A SOURCE (H g1 ,1 ) for all (t, v) [ 0, 1 ] × R, the function (s, x, y) g1 (t, s, x, y, v) is measurable; (H g1 ,2 ) there exists a nonnegative measurable function l1 : [ -, 0 ] × × satisfying 0 - q < + and such that ~ ~ |g1 (t, s, x, y, v) - g1 (t, s, x, y, v )| l1 (s, x, y) |t - t| + |v - v | ~ ~ ~ for all t, t [ 0, 1 ], v, v and for a.e. (s, x) [ -, 0 ] × ; ~ (H g1 ,3 ) the function t (-,0)× g1 (t, s, ·, y, 0) is bounded on [ 0, 1 ]. Of course, one could also consider some combinations of the two cases above, i.e. that some terms are as in Case 1 and some as in Case 2. However, for the sake of simplicity, we limit ourselves to dealing with Cases 1 and 2, only. Now we observe that in problem (6.1) A denotes the following secondorder linear symmetric differential operator in A= j,k=1 Dxj aj,k (x)Dxk - a0 (x). We assume that a0 C() and aj,k C 1 (), ai,j = aj,i , j, k = 1, . . . , n, satisfy a0 (x) > 0, x , j,k=1 aj,k (x)j k µ||2 , (x, ) × Rn , µ being a positive constant. Moreover, the operator B is defined by either of the following relations: (6.3) (6.4) (D) Bw(x) = w(x), x , (R) Bw(x) = DA w(x) + b0 (x)w(x), x , ALFREDO LORENZI and IOAN I. VRABIE where b0 C(), b0 (x) 0 for all x , and A is the conormal vector associated with A, i.e. (A (x))j = n j,k=1 i (x)aj,k (x), (x) denoting the outward normal vector at x . Here (D) and (R) stand, respectively, for the Dirichlet and the Robin boundary operators. Then, related to the reference space Xq = , q (1, +), we define D(Aq ) = w W 2,q ()) : Bw = 0 on , Aq w = Aw, w D(Aq ), the operator B being defined by (6.3) and (6.4). Then it is well-known (cf. e.g. Anikonov and Lorenzi [1]) that Aq generates in the semigroup {eAq t }t0 , which is analytic in the sector (/2)-p and satisfies the estimate eAq t e- , t (/2)-q . Moreover, from Vrabie [13, Proposition 2.2.1, p. 41]3 , taking into accounhat u := u + Au is a norm on W 2,q ()) equivaleno the usual one, and using the Sobolev, Rellich-Kondrachov Theorem see Vrabie [13, Theorem 1.3.8, p. 10], we deduce that (I - A) is a compact operator from into itself. Theorem 6.1. Lehe assumptions listed in Cases 1 and 2 be satisfied and let satisfy inequality (5.18). Then, the problem (IP1 ) has a unique mild solution (u, z) C 1 ((0, 1); ) C([0, T ]; D(A)) × , q (1, +), continuously depending on the data. Proof. We consider separately Case 1 and Case 2. We recall that, in both cases, X = . Case 1. The property (1 ) in (H ) immediately follows from the inequality in (H ), while property (2 ) is implied by the inequalities ~~ |(t, v)-(t, v )| t, (-,0)× ~ v(s, y) - t, (-,0)× v (s, y) ~ ~ l |t - t| + (-,0)× |(v - v )(s, y)| ~ C([0,1];) ~ l |t - t| + m() v - v ~ 3 According to Proposition 2.2.1, (I - A) is compact if and only if for each k > 0, the level set k = {u D(A); u + Au k} is relatively compact in X. IDENTIFICATION OF A SOURCE l being the Lipschitz constant related to . So, we have proved our assertion with = l max{1, m() }. Then, from the inequality ~~ |f0 (t, v)(x) - f0 (t, v )(x)|= g, x, 0 - ~ v(s, x) - g, x, v (s, y) ~ ~ l0 |t - t| + (-,0)× |(v - v )(s, x)| , ~ we get ~~ f0 (t, v) - f0 (t, v ) ) ~ l0 m() |t - t| + v - v ~ C([0,1];) Similarly, from the inequality ~~ |f1 (t, v)(x) - f1 (t, v )(x)| g1 t, x, y, ~ v(s, y) - g1 t, x, y, v (s, y) ~ ~ l1 (x, y) |t - t| + |(v - v )(s, y)| , ~ we obtain ~~ f1 (t, v) - f1 (t, v ) ~ |t - t| 0 |(v - v )(s, y)| ~ ~ |t - t| + v-v ~ 0 C([0,1];) q . From the previous estimate we easily deduce the property (f1 ) in (Hf ) with 1 = max l0 m() + , q . ALFREDO LORENZI and IOAN I. VRABIE Finally, from the definition of , we easily obtain the property (f2 ) with 2 = max g0 (·, ·, 0) C([ 0,1 ];) , g1 (t, ·, y, 0) Whence, we have checked that (H ) and (Hf ) are fulfilled. Case 2. We have to show that (H ) and (Hf ) are fulfilled. According to (H 2 ), the sub-condition (2 ) in (H ) is satisfied with mj = (-,0)× j (y) , = 1, 2, while the sub-condition (1 ) is implied from the following inequalities: ~~ |(t, v) - (t, v )| (-,0)× ~ |(t, s, y, v(s, y)) - (t, s, y, v (s, y))| ~ (-,0)× ~ l(s, y) |t - t| + |(v - v )(s, y)| ~ 0 (-,0)×L1 ()) ~ |t - t| l ~ |t - t| l (v - v )(s, ·) ~ . (-,0)×L1 ()) + v-v ~ C([-,0];) In this case we have = max (-,0)×L1 () Likewise, from the inequality ~~ |f0 (t, v)(x) - f0 (t, v )(x)| - 0 - 0 ~ |g0 (t, s, x, v(s, x)) - g0 (t, s, x, v (s, x))| ~ ~ l0 (s) |t - t| + |(v - v )(s, x)| , ~ we deduce ~~ f0 (t, v) - f0 (t, v ) ~ |t - t| l0 ~ |t - t| l0 0 L1 (-,0) m() + l0 (s) (v - v )(s, ·) ~ - L1 (-,0) L1 (-,0) m() + l0 v-v ~ C([-,0];) . IDENTIFICATION OF A SOURCE Similarly, from the inequality ~~ |f1 (t, v)(x) - f1 (t, v )(x)| (-,0)× ~ |g1 (t, x, y, v(s, y)) - g1 (t, x, y, v (s, y))| ~ ~ l1 (s, x, y) |t - t| + |(v - v )(s, y)| , ~ (-,0)× we get ~~ f1 (t, v) - f1 (t, v ) ~ |t - t| (-,0)× 0 |(v - v )(s, y)| ~ ~ |t - t| q v - v )(s, ·) ~ ~ |t - t| C([0,1];) 0 + v-v ~ q . Summing up, we have shown thahe sub-condition (f1 ) in (Hf ) hol true with 1 = max m() l0 l0 L1 (-,0) L1 (-,0) 0 , q . Finally, according to (H g0 ,3 ) and (H g1 ,4 ), the sub-condition (f2 ) in (Hf ) hol with any 2 = g0 (t, ·, ·, 0) L1 ((-,0);) L1 ((-,0);) , + g1 (t, ·, ·, y, 0) ALFREDO LORENZI and IOAN I. VRABIE as it easily follows from the formula f (t, v)(x) = g0 (t, s, x, 0) + (-,0)× g1 (t, s, x, y, 0) . Then, by virtue of Theorem 5.1 and we can conclude thahe problem (6.1), (6.2) admits a unique mild solution (u, z) continuously depending on the data with respeco the norms pointed out in the statement of Theorem 5.1. The proof is complete. 7. Application to a semilinear hyperbolic problem Finally, we give a simple example of an identification problem in which the operator A does not generate a compact semigroup but, nevertheless, k (I - A) is a compact operator. Let C , k = 1, 2, be the space of all k (R; R). -periodic functions u C 0 Let a R \ {0}, > 0, : [ 0, 1 ] × C([ -, 0 ]; C ) R+ be a Lipschitz 0 ) C 0 be a Lipschitz function, functional, f : [ 0, 1 ] × C([ -, 0 ]; C W 1,1 ([ 0, 1 ]; R), and let us consider the following identification problem for he transport functional equation with delay, in C : 0 0 (IP2 ) Let a R\{0} and > 0. Given h C([-, 0 ]; C ), 0 , C and 0 0 W 1,1 ([ 0, 1 ]; R) \ {0} , find z C and a mild solution u : [ 0, 1 ] C of the Cauchy Problem u (t, x) = -a u (t, x) - u(t, x) + (t, ut (·, x)))z(x) x t +f (t, ut (·, x)), (t, x) (0, 1) × Rn , u(t, x) = u(t, x + ), (t, x) (0, 1) × Rn , u(t, x) = h(t, x), t [ -, 0 ], x (0, ), satisfying the additional condition (t)u(t, x) dt = (x). 0 hroughout, we denote by · the usual -norm on C . From Theorem 3.1, we deduce heorem 7.1. Let > 0, : [ 0, 1 ] × C([ -, 0 ]; C ) R+ be a 0 0 C 1 -functional, f : [ 0, 1 ] × C([ -, 0 ]; C ) C be a C 1 -function and IDENTIFICATION OF A SOURCE W 1,1 ([ 0, 1 ]; R) \ {0} satisfying (t) 0 for each t [ 0, 1 ]. Let us assume thahere exist > 0, 1 > 0, 2 > 0, m1 > 0 and m2 > 0 such that (7.1) (7.2) (7.3) |(t, v) - (t, v)| |t - t| + v - v m1 (t, v) m2 , f (t, v) - f (t, v) 0 C([ -,0 ];C ) 1 |t - t| + v - v 0 C([ -,0 ];C ) 0 for each (t, v), (t, v) [ 0, 1 ] × C([ -, 0 ]; C ) and (7.4) f (t, 0) 1 0 for each t [ 0, 1 ]. Let h C([ -, 0 ]; C ) with h(0) C and let us assume that (3.1) hol true. Then, there exists a mild solution (z, u) to the problem (IP2 ), admitting the implicit representation given by u(t, x) = h(t, x) for t [ -, 0 ] and x R, u(t, x) = e-t h(0, x - at) + t e-(t-s) f (s, us (·, x - a(t - s))) e-(t-s) (u(s, us (·, x - a(t - s)))zs (x - a(t - s)) , where z = F (u), F being defined by (2.5) and (2.6). Let us assume further that > 1 is sufficiently large as to satisfy (7.5) and (7.6) > max {(m2 L0 ), r2 (, h) + m2 r3 (, h) }, 1 - e- < min {(m2 L0 ) , r2 (, h) + m2 r3 (, h) where r2 (, h) is given by (5.11) and r3 (, h) is given by (5.15). Then, the mild solution above is unique and the application (, h) (z, u) from 1 0 0 0 C × C0,A ([ 0, 1 ]; C ) to C × C([ 0, 1 ]; C ) is locally Lipschitz. Here A : 1 0 0 1 C C C is defined by Au = -au - u, for each u C . ALFREDO LORENZI and IOAN I. VRABIE Proof. Let us observe that (IP3 ) can be equivalently rewritten as (IP1 ) 0 in X = C , where A, , f , are as above. Clearly A = A1 - I, where 1 A1 u = -au for each u C . By Vrabie [14, Problem 3.4, p. 74], we know hat A1 generates a C0 -group of isometries on C , {T (t); t R}, defined by [T (t)](x) = (x - at) 0 for each C , each x R and each t R. Thus A generates a C0 0 semigroup of contractions on C , {S(t); t 0}. One may easily verify that S(t) = e- (t) for each t 0. Moreover, by the infinite dimensional version of Arzel`a Ascoli's Theorem see Vrabie [13, Theorem 1.3.1, p. 7] it follows that for each k > 0, the set {u D(A); u + Au k} 0 is relatively compact in C . By Vrabie [13, Proposition 2.2.1, p. 41], we is compact. So, from this remark and (7.1) (7.4), deduce that (I - A) it follows that all the hypotheses of Theorem 3.1 are satisfied and this completes the proof of the existence part of Theorem 7.1. For the uniqueness and continuous dependence part, we have only to observe that, in our case, S(·) L1 1 - e- and thus, we are in the hypotheses of Theorem 5.1, wherefrom the conclusion.

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

**Published: ** Jan 1, 2015

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