Determination of initial data for a reaction-diffusion system with variable coefficients

In this paper, we study a final value problem for a reaction-diffusion system with time and space dependent diffusion coefficients. In general, the inverse problem of identifying the initial data is not well-posed, and herein the Hadamard-instability occurs. Applying a new version of a modified quasi-reversibility method, we propose a stable approximate (regularized) problem. The existence, uniqueness and stability of the corresponding regularized problem are obtained. Furthermore, we also investigate the error estimate and show that the approximate solution converges to the exact solution in \begin{document}$L_2$\end{document} and \begin{document}$\stackrel{0}{H_1}$\end{document} norms. Our method can be applied to some concrete models that arise in biology, chemical engineering, etc.

1. Introduction. In this paper, we consider the question of finding the functions u k (x, t), (x, t) ∈ D T , satisfying the problem with the final value condition u k (x, T ) = ϕ k (x), in Ω, k = 1, m.
Although the initial value problem for (1) is classical and has been so studied, however, the study of final value problem (called Initial inverse problem) for (1) is limited and open. The properties of the initial inverse problems are very different from the direct problem. For the direct problem, we often investigate the properties of the solution such as the existence, the blow up, the decay, etc. To our knowledege, the existence and uniqueness of solution of (1) is an open problem, and we do not investigate this problem here. Otherwise, our main purpose is described as follows. We assume that Problem (1)-(2) has a unique solution in a suitable space and we will investigate its approximation. In practice, we get the data ϕ k , k = 1, m by observed data. Hence, instead of ϕ k , k = 1, m, we shall get an approximate data ϕ k , k = 1, m satisfying where the constant > 0 represents a bound on the measurement error. In fact, from a small perturbation of a physical measurement, the corresponding solution may have a large error. This makes the numerical computation delicate. Hence a regularization is needed. Now we dwell on backward problems: • For m = 1, many papers are devoted to special cases of (1) restricted to η = 1 or η = η(t), for example, [5,13,20] and references therein. As for the backward problem of a general parabolic equation in the case of η = η(x, t), less study has been done. • For m = 2, the first result on the backward problem for a system of parabolic equations is presented by P. Schaefer [19]. However, the regularization method with error estimate has not been studied in [19]. (2) is not difficult since it can be transformed to nonlinear integral equations. In this case, we follow spectral regularized method for solving it, such as nonlinear integral equation, truncation method, modified integral methods, quasiboundary value method, etc. (See [13] and the references therein). However, with general form of η(x, t), the system (1)- (2) can not be transformed into a nonlinear integral equation as in the Fourier series. Hence, some classical methods and previous techniques are not applicable to approximate system (1). This case is more difficult for investigation and a new method is required. For the inverse problem, we assume that the solution of the system (1) exists. In this case its solution is not stable. In this paper, we propose a new modified quasi-reversibility method to regularize system (1) in case of global or local Lipschitz function F. The method of classical quasi-reversibility has now a quite long history since the pioneering book of Lattes and Lions [12]. The original idea of these authors was, starting from an ill-posed problem which satisfies the uniqueness property, to introduce a perturbation of such problem involving a small positive parameter . In [12], the authors considered a linear homogeneous backward parabolic problem with space and time dependent coefficients in the following form A(t) = −∇ · (η(x, t)∇·) and they suggested a regularized problem by perturbing directly A(t) by A(t)+βA * (t) in order to obtain the following approximate problem and their results did not give a convergence rate. At first, we intend to apply the method of [12] for solving the system (1). We can show that the approximate problem is well-posed but we can not show that the convergence rates tend to zero. In this paper, we will not approximate directly the operator ∇·(η(x, t)∇·) as in [12]. We emphasize that our method is original and very different from the methods in [12]. The main idea of the paper is of approximating := R∆, R > 0, by a bounded operator, in order to establish an approximation for the regularized problem. By using our new tool, we show that the regularized problem is well-posed; then we estimate the error between a solution of (1)-(2) and the approximate solution. The paper is organized as follows. Section 2 introduces some preliminaries and notations and shows the ill-posedness of the system (1). In Section 3, a stability estimate is proved under an a priori condition on the exact solution and the globally Lipschitz source term. In Section 4, the analysis is extended to local Lipschitz source functions and perturbed time dependent coefficient. To show the well-posedness of the regularized problem, we apply the Faedo-Galerkin and the compactness method. We propose a new idea in which the locally Lipschitz source functions F k , (k = 1, m) are approximated by a sequence F k ( ) of globally Lipschitzian functions. Finally, in section 5, we introduce some specific systems which can be tackled by our method.
We introduce the abstract Gevrey class of functions of index δ > 0, see, e.g. [3], defined by which is a Hilbert space when equipped with the inner product , for all u, v ∈ V δ (Ω); its corresponding norm is as follows Now, we return to the study of the ill-posedness of Problem (1). For sake of simplicity, in (1), we assume that η k (x, t) = η k (t) ≥ r > 0, ∀t ∈ [0, T ], k = 1, m. For any p ∈ N * , let the final data ϕ (p) k , k = 1, m be as follows and the source function It is easy to see that the solution of problem (1) is then given by (for k = 1, m) Step 1. The existence and uniqueness of the solution of the system (9).
At first, we show that (9) has a unique solution u for k = 1, m and Let any (w 1 , w 2 , ..., w m ) ∈ [C([0, T ]; L 2 (Ω))] m , using Hölder inequality, we have The latter inequality holds for all k = 1, m. Thus, we obtain Hence J is a contraction. Using the Banach fixed-point theorem, we conclude that has a unique solution Step 2. The instability of the solution of the system (1). The norm of B 1 in L 2 (Ω) is estimated as follows The norm of B 2 in L 2 (Ω) is bounded as follows .
By computations analogous to the previous one for k = 1, m and summing up the obtained results, one has .
As p → +∞, it follows from (8) and (15) that Thus, the problem (1) is, in general, ill-posed in the Hadamard sense.
3. The inverse problem with global Lipschitz reaction terms. In this section, we present a regularized problem for approximating the system (1)- (2). We now make the following assumptions: (A 1 ) Let r, R be positive constants. Suppose η k : D T → R, is a continuous function such that with for any function w ∈ L 2 (Ω). The function β := β( ) is such that lim →0 + β = 0; it plays the role of a regularization parameter.

VO VAN AU, MOKHTAR KIRANE AND NGUYEN HUY TUAN
Using the modified quasi-reversibility method, we introduce the following regularized problem where k = 1, m and ϕ k ∈ L 2 (Ω) satisfies (4). The main tool for our proofs will be the following lemma.
2. For any w ∈ L 2 (Ω), it holds Proof. Using the inequality log(1 + a) ≤ a, for any a > 0, we may estimate From (18) and (19), it is easy to calculate that Using Parseval's equality, we deduce that The proof is complete.
Step 1. The existence and uniqueness of the solution to the regularized system (20).

VO VAN AU, MOKHTAR KIRANE AND NGUYEN HUY TUAN
This implies that Z ,β k satisfies the following system where the functions G k (k = 1, m), are defined by for any w k ∈ C ([0, T ]; L 2 (Ω)). Put In the space H 1 (Ω), we take a basis {e j } ∞ j=1 and define the finite dimensional space V p = span{e 1 , e 2 , ...e p }.
Let ϕ k,p , k = 1, m be an element of V p such that as p → +∞. We search the approximate solution of the problem (30) in the form where the coefficients c k, pj (t) satisfy the system of linear differential equations for any u, v ∈ H 1 (Ω). The existence of a local solution of system (34) is guaranteed by Peano's theorem [14].
Multiplying the jth equation of (34) by c k, pj (t) and summing up with respect to j, we obtain By integrating (36) from t to T , we obtain The assumption ρ k (x, t) ≥ (R − r) allows to write Using Hölder inequality and Lemma 3.1, we deduce that Now we estimate the first term of (40). First, using (17), we get This implies that dµ. (44) Since (32), we deduce that where C k ( ) is a constant independent of p. By summing up the obtained results (44), one has Let us set dµ, then we know from (46) that By using Gronwall's inequality, we obtain that for all p ∈ N. This implies that and Z ,β k,p is bounded in L 2 (0, T ; H 1 (Ω)), k = 1, m, Then we can extend the solution to the interval [0, T]. Now we need to pass to the limit when p → +∞. Since ∇ · k (x, t)∇Z ,β k,p defines an element of H −1 (Ω), it is easy to see that Due to the Banach-Alaoglu theorem, from (49), (50) and (51), we can extract subsequences which we denote by the same symbol Z ,β k,p such that when p → +∞ and k = 1, m. Using Aubin?Lions compactness lemma, we obtain Hence, passing if necessary to a subsequence, still denoted by Z ,β k,p , one has Since F k is globally Lipschitz, we also obtain Passing to the limit in (34) by (52), (53), (55), we have Z ,β k ∈ L ∞ (0, T ; L 2 (Ω)) ∩ L 2 (0, T ; H 1 (Ω)), satisfying the system (30). Therefore, we can conclude that the system (20) has a unique solution U ,β such that Step 2. The uniqueness of solutions of (20).
where q β > 0 depends of β to be chosen later. It is easy to see that
where q β > 0 depends of β to be chosen later. Then, from the equalities (66) and (67), a simple computation gives for all (x, t) ∈ D T and

VO VAN AU, MOKHTAR KIRANE AND NGUYEN HUY TUAN
First, thanks to Hölder's inequality, we have the estimate Second, using log(1 + a) ≤ a, for any positive a, the term II can be estimated as follows where we have used the fact that exp (2q β (t − T )) ≤ 1, ∀t ∈ [0, T ] and Finally, using (17), we estimate Combining (70)-(72) and (74) yields Consequently, This implies that By computations analogous to the previous one for k = 1, m and summing up the obtained results, one has where we have used the inequality ( ; choosing q β = 1 T log 1 β > 0 and using (4) and (26), we observe that Gronwall's inequality allows to obtain This implies that Recalling By putting we get which leads to (27).
To obtain the approximation of the solution U at t = 0, we choose a number t ∈]0, T [ such that lim →0 + t = 0. Then we see that U ,β (x, t ) is an approximation of U(x, 0). Indeed, (82) implies that where we define U t = (∂ t u 1 , ..., ∂ t u m ).
It is easy to show that for every β > 0, there exists a unique t ∈]0, T [ such that lim →0 t = 0 and t = β t T . This implies that log t t = 1 T log β. Using the inequality log t > − 1 t for all t > 0, we obtain that t < T log 1 β ; it is straightforward to see then that (using (26)) therein, without loss of generality, we may assume that mE ≤ C(α, β, m, E, T ) and using the fact that e mKT +T /2 ≥ 1 then, the estimate (28) follows. This completes the proof of the theorem.
4. The inverse problem with locally Lipschitz reaction. Section 2 has addressed a problem in which F is a globally Lipschitz function, in this section we extend the analysis to a locally Lipschitz function F. Results for the locally Lipschitz case are still very scarce. Hence, we have to find another regularization method to study the problem with the locally Lipschitz source which is similar to the latter source.
For emphasis and clarity, we outline our procedure. Keeping (A 1 ), we replace the global Lipschitz assumption (A 2 ) by the local Lipschitz (A 4 ), specifically: where (x, t) ∈ D T and We note that K( ) is increasing and lim →+∞ K( ) = +∞. Now, we outline our ideas to construct a regularization. For all > 0, we approximate F by F( ) defined by We still pay more attention to the system (20) from replacing η k by η k that satisfies (86). We are going to introduce the main idea to solve this problem with a special generalized case of source term defined by (88), we consider the problem: where k = 1, m and for each > 0, we consider a parameter → +∞ as → 0 + . Before stating the main theorem, we first consider the following lemma.
given as (88). Then we have Proof. First, we show that for any u k , v k ∈ R then To prove (91) , we divide all following cases.
Now we return to the proof of Lemma (4.1). Indeed, since the fact that u k , v k ≤ l and using (87), we have where we have used (91) for the last estimate.

4.1.
Error estimate in L 2 -norm. We then have the convergence result given by the following theorem. 2. If we choose γ > 0 satisfying 0 < γ < m −1 2 then the right-hand side of (96) converges to zero, as → 0 + .
Proof. First, we note that the proof of the existence and uniqueness of the solutions to the problem (1) is completely the same as in Step 1 of Theorem 3.2. We pass to the error estimate between the regularized solution of system (90) and the exact solution of system (1).
For (x, t) ∈ D T , we begin by establishing functions k (x, t), k (x, t) that satisfy r ≤ k (x, t), k (x, t) ≤ R, k = 1, m, such that k (x, t) By an argument analogous to that used for the proof of Theorem 3.2, we infer that (k = 1, m) First, estimating X 1 and X 2 is totally similar to (71) and (72) respectively. Next, since lim →0 + = +∞, for a sufficiently small > 0, there is an > 0 such that ≥ u k L∞(0,T ;L2(Ω)) , k = 1, m. For this value of , we have .