Reconstruction of a Source Term in a Parabolic Integro-Differential Equation from Final Data

Heat flow processes in media with memory are governed by parabolic integrodifferential equations [7]. A number of papers is devoted to inverse problems to determine kernels of these equations in different formulations making use of measurements over time (see e.g. [4, 6, 7, 8, 11, 13, 14]). Recently some papers appeared that deal with the reconstruction of source terms or coefficients of these equations making use of final or integral overdetermination [5, 12]. In particular, the authors’ paper [5] extends former existence and uniqueness results of Isakov [3] to the integro-differential case. The existence of the solutions to the inverse problems to determine unknown source terms from final over-determination of the temperature requires sufficient regularity and a certain monotonicity of a time-component of this term. In the present paper we follow another approach. Instead of the conventional solution, we deal with the quasi-solution of the inverse problem that uses final data. Then we can build up a theory without any smoothness or monotonicity restrictions on the source. Similar results in the case of the parabolic differential equation without an integral term in the one-dimensional case were obtained by Hasanov [2]. Quasi-solutions of other integro-differential inverse problems were studied in [1, 9].


Introduction
Heat flow processes in media with memory are governed by parabolic integrodifferential equations [7]. A number of papers is devoted to inverse problems to determine kernels of these equations in different formulations making use of measurements over time (see e.g. [4,6,7,8,11,13,14]).
Recently some papers appeared that deal with the reconstruction of source terms or coefficients of these equations making use of final or integral overdetermination [5,12]. In particular, the authors' paper [5] extends former existence and uniqueness results of Isakov [3] to the integro-differential case. The existence of the solutions to the inverse problems to determine unknown source terms from final over-determination of the temperature requires sufficient regularity and a certain monotonicity of a time-component of this term.
In the present paper we follow another approach. Instead of the conventional solution, we deal with the quasi-solution of the inverse problem that uses final data. Then we can build up a theory without any smoothness or monotonicity restrictions on the source. Similar results in the case of the parabolic differential equation without an integral term in the one-dimensional case were obtained by Hasanov [2]. Quasi-solutions of other integro-differential inverse problems were studied in [1,9].

Direct Problem
Let Ω be a n-dimensional domain with sufficiently smooth boundary Γ and Γ = Γ 1 ∪ Γ 2 where meas Γ 1 ∩ Γ 2 = 0. Assume that for any j ∈ {1; 2} it holds either Γ j = ∅ or meas Γ j > 0. Denote Ω T = Ω × (0, T ), Γ 1,T = Γ 1 × (0, T ), Γ 2,T = Γ 2 × (0, T ). Consider the problem (direct problem) to find u(x, t) : The problem (2.1)-(2.4) describes the heat flow in a body Ω with the thermal memory. Concerning the physical background we refer the reader to [7]. The solution u is the temperature of the body and m is the heat flux relaxation (or memory) kernel. The boundary condition (2.4) is of the third kind where the term −ν A · ∇u + m * ν A · ∇u equals the heat flux in the direction of the co-normal vector.
Let us introduce some additional notation. Let X be a Banach space. We denote by C( In addition, we need spaces of fractional order and anisotropic spaces. To this end, let us first introduce the following notation for difference quotients of xand (x, t)-dependent functions with powers: where |x| denotes the Euclidean norm of x in the space R n . For any l ≥ 0 we introduce the Sobolev-Slobodeckij spaces (cf. [10,15]) and integrate by parts with respect to time and space variables. We obtain the following relation: This relation makes sense also in a more general case when φ satisfies only (2.7) and u doesn't have regular first order time and second order spatial derivatives. We call a weak solution of the problem (2.1)-(2.4) a function from the space that satisfies the relation (2.9) for any η ∈ T (Ω T ) and in case Γ 1 = ∅ fulfills the boundary condition (2.3).
Proof. It is well known (see e.g. [10]) that in the particular case m = 0 the solution exists, is unique and the operator H, that assigns to the data vector u 0 , g, h, f , φ the weak solution is Lipschitz-continuous from the space L 2 (Ω)×W 1 2 , 1 4 2 (Γ 1,T )×L 2 (Γ 2,T )×L 2 (Ω T ) n+1 to the space U(Ω T ). Let us denote G(f, φ) = H(0, 0, 0, f, φ). Then, denoting by u the solution corresponding to m = 0, the problem (2.1)-(2.4) for u is in U(Ω T ) equivalent to the following operator equation for the function v = u − u: with the linear operator F v = G(−m * (av), −m * ( n j=1 a ij v xj )). We are going to estimate F . To this end, we make use of the following inequality that immediately follows from the estimate (19) in [5]: (2.11) Here Ω t = Ω × (0, t) for t ∈ (0, T ) and w is an arbitrary element of L 2 (Ω T ). Moreover, we define the cutting operator P t by the formula Note that it holds G(P t f, P t φ)(x, t) = G(f, φ)(x, t) for any (x, t) ∈ Ω t . Therefore, observing the Lipschitz-continuity of G and (2.11) we can estimate as follows: for any t ∈ (0, T ) with some constants C 1 , C 2 . Now we introduce the weighted norms in U(Ω T ): v σ = sup 0<t<T e −σt v U (Ωt) where σ ≥ 0. Using the deduced estimate for F we obtain

Since
T 0 e −σs |m(s)| ds → 0 as σ → ∞, the operator F is a contraction for sufficiently large σ. Consequently, (2.10) has a unique solution in U(Ω T ). This proves the existence of the unique weak solution of (2.1)-(2.4).
Secondly, let us prove the classical solvability assertion of the theorem. Again, we use the results in case m = 0. It is known [15] that in case m = 0 the solution belongs to W 1 2 (Ω T ) and the operator H 1 that assigns to the data vector u 0 , g, h, f the classical solution is Lipschitz-continuous from the space where . This time we have to introduce a more complicated extension operator instead of P t because the argument of F 1 has traces on slices Ω × {t}. Let us define for s > 2t in case 2t < T.
Then, since the function v in the range of F 1 satisfies v| t=0 = 0, it holds P t v ∈ W 2,1 2 (Ω T ) for t ∈ (0, T ). Moreover, , where h = m * ν A · ∇v| Γ2,T and f = m * Av. Consequently, in view of the Lipschitz-continuity of G 1 we deduce for any t ∈ (0, T ) with some constant C 3 and Γ 2,t = Γ 2 × (0, t). Using the trace theorem for Sobolev-Slobodeckij spaces [10] and the relation (m * v) t = m * v t , that holds due to v| t=0 = 0, we compute with some constant C 4 . Applying this estimate in (2.13) and using (2.11) we deduce in the space W 2,1 2 (Ω T ) and, as in the first part of the proof, show that F 1 is a contraction in W 2,1 2 (Ω T ) if σ is sufficiently large. This proves the unique solvability of (2.12) and in turn the classical solvability assertion of theorem.

Formulation of Inverse Problem. Existence of Quasi-Solution
Let F be a linear closed subspace of L 2 (Ω T ). Suppose that the source term f is of the following form: We pose an inverse problem to determine the function F ∈ F making use of the final measurement More precisely, we will search a quasi-solution of this problem. This is a solution of the following minimization problem for the cost functional: find where F ⊆ F is a subset including constraints. Here u(x, t; F ) stands for the solution of the direct problem corresponding to the given F . Let us introduce some cases of F .
Let Ω be a cylinder: ..,N ∈ (L 2 (Ω T )) N , κ = 0 is a prescribed vectorfunction. In practice, the component κ j may be the characteristic function of a subdomain Ω j ⊂ Ω.

Now let us consider the first variation of the cost functional
where ∆u(x, t; F ) = u(x, t; F + ∆F ) − u(x, t; F ). By Theorem 1, the function ∆u belongs to W 2,1 2 (Ω T ) and solves the following problem in the classical sense: Moreover, let us introduce the following adjoint problem with the solution ψ(x, t; F ): It is easy to see that the equivalent problem forũ(x, t) = ψ(x, T − t; F ) is of the form (2.1)-(2.4) with homogeneous differential equation and boundary conditions and the initial conditionũ Therefore, applying Theorem 1 we conclude that problem (3.7)-(3.10) has a unique weak solution. The weak problem for ψ(x, T − t; F ) reads Lemma 1. It holds the following formula: Proof. Since ∆u ∈ W 2,1 2 (Ω T ) satisfies the homogeneous boundary condition on Γ 1 , it holds ∆u(x, T − t, F ) ∈ T (Ω T ). Let us use the test function η(x, t) = ∆u(x, T − t, F ) in (3.11). This yields (changing the variable t by T − t under the integrals and observing that η(x, T ) = 0 and omitting F in the arguments for the sake of shortness) On the other hand, the problem (3.3)-(3.6) in the weak form reads Since ∆u ∈ W 2,1 2 (Ω T ) has the regular time derivative, we can integrate by parts the integral ΩT ∆uζ t dx dt in (3.14). This results in the relation It is important that this relation doesn't contain the time derivative of the test function ζ. Therefore, we can extend the set of test functions of (3.15) from In particular, it is possible to take the test function ζ = ψ ∈ U 0 (Ω T ). Then we obtain Subtracting (3.16) from (3.13) and changing the order of integration in convolution terms we deduce the formula (3.12). Lemma is proved.
Theorem 2. Let F be a bounded, closed and convex subset of F . Then the problem (3.1) has a solution in F . Moreover, the set of all solutions F * form a closed convex subset of F .
Proof. The assertion follows from Weierstrass existence theorem (see [16, Section 2.5]) once we have proved that J(F ) is weakly sequentially lower semicontinuous in F , i.e.
Let us compute: is the change of u corresponding to the change of the free term ∆F n = F n − F . Thus, in view of (3.12) we have Since ψ ∈ L 2 (Ω T ), this implies the relation (3.17). To prove the convexity, we firstly note that Therefore, in view of the convexity of the quadratic function we obtain This shows the convexity of J. Theorem is proved.

Remark 1.
In order to prove the existence in an unbounded set F incl. F , it is sufficient to have the weak coercivity of J(F ). This is a difficult problem, because monotonicity methods in general fail for problems in integro-differential PDE. However, the boundedness assumption of F seems not very restrictive, because in practice some bound for F may be available.

Regularized Problem
In [5] we proved that in a particular case the solution of the inverse problem under consideration continuously depends on certain derivatives of the data.
This shows the ill-posedness of the problem in case the data have noise in L 2 space. We can easily incorporate Tikhonov regularization in quasi-solution. In this case we minimize the stabilized cost functional: find Here α > 0 is the regularization parameter that depends on the noise level of the data u T . If we set here α = 0, we get the original problem (3.1).
Theorem 3. Let α > 0 and F be a closed and convex subset of F (may be also F = F ). Then the problem (4.1) has a unique solution in F .
Proof. Obviously the additional term I(F ) = α F L 2 (ΩT ) is strictly convex: and weakly coercive, i.e., I(F ) → ∞ as F L 2 (ΩT ) → ∞. This makes the whole functional J α strictly convex and weakly coercive. Moreover, it is easy to check that I(F ) is weakly sequentially lower semi-continuous. Since is also weakly lower semi-continuous (this was shown in the proof of Theorem 2), the whole functional J α is weakly lower semicontinuous. Now the assertion of the theorem follows from Weierstrass existence theorem [16, Section 2.5].

Auxiliary Estimates
Lemma 2. The following estimate is valid with a constant C 0 : ∆u(·, T ; F ) L 2 (Ω) ≤ C 0 ∆F L 2 (ΩT ) . (5.1) Proof. For the sake of shortness, we omit F in the list of arguments of ∆u. Firstly, we prove this assertion in case m L 1 (0,T ) is small enough and the equation for ∆u (3.3) contains an additional term, namely it has the form where σ is a sufficiently large number such that σ − a(x) ≥ for any x ∈ Ω. By Theorem 1, ∆u belongs to W 2,1 2 (Ω T ) and solves the problem (5. Further, we use the inequalities ∆u xi L 2 (ΩT ) ≤ |∇∆u| L 2 (ΩT ) , i = 1, . . . , n, and definition of I (see (5.4)). We have Therefore, in case m satisfies the smallness condition we obtain I 2 ≤ 2 √ ∆F L 2 (ΩT ) I that yields I ≤ 2 √ ∆F L 2 (ΩT ) . Observing that ∆u(·, T ) L 2 (Ω) ≤ √ 2I, from the latter inequality we deduce the estimate (5.1) with the constant C 0 = 2 √ 2/ . Now let us return to the original problem (3.3)-(3.6) without the additional σ-term and arbitrarily large m. Define the following function: ∆u σ (x, t) = e −σt ∆u(x, t) where σ ∈ R. It is easy to check that ∆u σ solves the following problem: where m σ (t) = e −σt m(t) and ∆F σ (x, t) = e −σt ∆F (x, t). Clearly, there exists a sufficiently large σ such that m σ satisfies the condition (5.7) and the inequality σ − a(x) ≥ is valid for x ∈ Ω. Therefore, the first part of the proof applies to the function ∆u σ . This means that the estimate is valid. Finally, in view of ∆u σ (x, T ) = e −σT ∆u(x, T ) and |∆F σ (x, t)| ≤ |∆F (x, t)|, from (5.8) we obtain the desired estimate (5.1) with the constant C 0 = 2 √ 2e σT / . Lemma 2 is proved.

Lemma 3.
The following estimate is valid with a constant C 1 : Proof. Proof is similar to the proof of the previous lemma. Observing (3.7)-(3.10) we see that the problem for ∆ψ(x, t; F ) has the following form: We start by proving the assertion in case m L 1 (0,T ) is small enough and the equation (3.3) contains an additional term, namely it has the form where σ is again sufficiently large, i.e. σ − a(x) ≥ for any x ∈ Ω. Since ∆u ∈ W 2,1 2 (Ω T ), by the trace theorem it holds ∆u| t=T ∈ H 1 (Ω). Moreover, one can immediately check that the time-inverted function ∆ψ(x, T − t; F ) satisfies a problem of the form (2.1)-(2.4) with an homogeneous equation, homogeneous boundary conditions and the initial condition 2∆u(x, T ; F ). Therefore, applying Theorem 1 we see that the function ∆ψ(x, t; F ) belongs to W 2,1 2 (Ω T ) and satisfies the problem (5.14), (5.11), (5.12), (5.13) in the classical sense. For the sake of shortness we omit the argument F of ∆ψ and ∆u in forthcoming computations. Multiplying (5.14) by ∆ψ and integrating by parts we obtain Observing the final condition (5.11) and rearranging the terms we get The left-hand side of (5.15) is estimated from below: a ij ∆ψ xj ∆ψ xi For the right-hand side of (5.15) we use the Cauchy-Schwarz inequality: = m * v L 2 (ΩT ) for any v.

Frechet Derivative and Gradient Method
It follows from Lemma 2 with (3.2) that the functional J is Frechet differentiable in L 2 (Ω T ). Moreover, according to Lemma 1, J (F ) is identical to the element ψ(F ) = ψ(x, t; F ) in L 2 (Ω T ), i.e. it holds Similarly, J α is Frechet differentiable in L 2 (Ω T ) and Therefore, gradient-type methods can be used to solve the minimization problems (3.1) and (4.1). These methods must be combined by proper projection techniques to get minimum in the subset F . However, it is possible to simplify the minimization procedure in case the structure of the subspace F is simple.
where L α = (2α + C 1 ) κ 2 L 2 (0,T ) . The cases 2 and 3 can be treated in a similar manner. Let us summarize the results in these cases.
The problem (4.1) admits the following form: find w * = arg min w∈W3 Φ 3,α (w). The functional Φ 3,α is Frechet differentiable, Φ 3,α (w) is identical to the element ( ΩT [2α N l=1 w l κ l (x, t) + ψ(x, t, N l=1 w l κ l )]κ j (x, t) dx dt) j=1,...,N of R N and the estimate is valid. In the following, let Φ α be one of the functionals Φ j,α , j = 1, 2, 3, defined above and W be the corresponding set of admissible solutions W j . Then we consider the problem find w * = arg min w∈W Φ α (w). (6.6) For the sake of simplicity, we assume that F = F . This means that we consider the unconstrained minimization and W is L 2 (Ω), L 2 (S T ) and R N in the cases 1, 2 and 3, respectively. Let w 0 ∈ W be an initial guess and compute the successive approximations by means of the gradient method with steps c k > 0. Let us perform a little analysis for this iteration process following partially the example of [2].
Clearly, the highest decrease rate of Φ α (w k ) is achieved in case c k = 1/L α when q k has the biggest value q k = 1/2L α . Theorem 5. Let α > 0 and c k be chosen as in Theorem 4. Then the sequence w k strongly converges to the unique solution of the minimization problem (6.6).
Proof. The existence of the unique solution for the minimization problem immediately follows from Theorem 3 and the definitions of Φ α . Moreover, since J α is weakly sequentially lower semi-continuous, strictly convex and weakly coercive (see the proof of Theorem 3), the same properties are valid also for Φ α . It is well-known that under such properties every minimizing sequence of Φ α weakly converges to the minimum point w * . Thus, firstly, let us show that w k is a minimizing sequence, i.e. Φ α (w k ) → Φ α (w * ).
Note that the sequence w k is bounded. Indeed, otherwise there exists a subsequence w ki such that w ki → ∞ and by the weak coercitivity it holds Φ α (w ki ) → ∞ which contradicts to the statement of Theorem 4 that Φ α (w k ) is monotonically decreasing.