SEMIDISCRETIZATION FOR A NONLOCAL PARABOLIC PROBLEM

A time discretization technique by Euler forward scheme is proposed to deal with a nonlocal parabolic problem. Existence and uniqueness of the approximate solution are proved.


Introduction
In this work, we study the time discretization by Euler forward scheme of the nonlocal initial boundary value problem with Ω ⊂ R d (d ≥ 1) a bounded regular domain and λ a positive parameter. The hypotheses we will assume on f are the same as in [6]. We recall first that (1.1) arises by reducing the following system of two equations modeling the thermistor problem: 2. The semidiscrete problem 2.1. Existence and uniqueness. We consider the Euler scheme (1.3), with Nτ = T, T > 0 fixed, and 1 ≤ n ≤ N, under the following hypotheses.
(H2) There exist positive constants σ,c 1 ,c 2 , and α such that α < 4/(d − 2) and for all ξ ∈ R, In the sequel, we will denote the norms in the spaces L ∞ (Ω), L k (Ω) by | · | L ∞ (Ω) and | · | k , respectively, (·,·) will denote the associated inner product in L 2 (Ω) or the duality product between H 1 0 (Ω) and its dual H −1 (Ω). Theorem 2.1. Let (H1)-(H2) be satisfied. Then, for each n, there exists a unique solution in Ω, (x,v) in Ω, . For a fixed v ∈ H 1 0 (Ω), (2.3) has a unique solution U ∈ H 1 0 (Ω). Then, for each µ ∈ [0,1], the operator S(µ,·) is well defined. Moreover, S(µ,·) is compact from H 1 0 (Ω) into it self. Indeed, using (H2), we have the estimate We can easily see that µ → S(µ,v) is continuous and that S(0,v) = U, for any v, if and only if U = 0. From Leray-Schauder fixed point theorem, there exists therefore a fixed point U of S(µ,·). Now, we derive an a priori estimate. Proof. The proof is similar to the one used by De Thélin in [10] concerning a very different problem and we will give here only a sketch. Suppose that d ≥ 2 and define (2.5) For each k ∈ N * , we consider the number we have Lemma 2.3. For all k ∈ N * , U n ∈ L qk (Ω), and moreover Proof. We prove by recurrence that U ∈ L qk . The property is true for 2) by |U| qm−γ U, using (H2), and Young's inequality, we get On the other hand, we have The rest of the proof follows the same lines as in [10, pages 383-384].
Uniqueness. Consider U and V two different solutions of (2.2) and define w = U − V . Then, we have (2.13) Multiplying (2.13) by w, integrating on Ω, and using the L ∞ -estimate obtained in Lemma 2.2, we get (2.14) Therefore, w = 0 if τ ≤ 1/c 9 .
We address now the question of stability.

using Lemma 2.2, and
Hölder's inequality, we obtain after simplification By induction and taking the limit in the resulting inequality as m → +∞, we get (ii) Multiplying the first equation of (1.3) by U k and using the hypotheses on f , one easily has A. El Hachimi and M. R. Sidi Ammi 1659 Using the elementary identity 2a(a − b) = a 2 − b 2 + (a − b) 2 and summing from k = 1 to n, we obtain Then, the inequalities (b)-(c) hold by using the uniform bound of U n in L ∞ which is established in part (a).

Error estimates for solutions
We will adopt the following notations concerning the time discretization for problem (1.1). We denote the time step τ = T/N, t n = nτ, and I n = (t n ,t n−1 ) for n = 1,...,N. If z is a continuous function (resp., summable), defined in (0, T) with values in H −1 (Ω) or L 2 (Ω) or H 1 0 (Ω), we define z n = z(t n ,·), z n = (1/τ) In z(t,·)dt, z 0 = z 0 = z(0,·); the error e n = u(t) − U n for all t ∈ I n and the local errors e n u and e n defined by e n u = u n (t) − U n , e n = u n − U n .
We have the following theorem.     We now estimate I 1 22 . Using the boundedness of ∂u/∂s (see [6]), we have  In the same manner, we have (4.11) Next, we estimate the first term on the right-hand side of (4.5) by using Hölder's and Young's inequalities and (H1),       The last inequality follows by using simultaneously the L ∞ -estimate of u(t) (see [6]), U n , and the error bound given in (a). Arguing exactly as in the previous estimate, we get