A bstract SUPERSOLUTIONS AND STABILIZATION OF THE SOLUTION OF A NONLINEAR PARABOLIC SYSTEM

Let us consider a nonlinear parabolic system of the following type: [la formula corresponent es troba al document] with Dirichlet boundary conditions and initial data. In this paper, we construct sub-supersolutions of (S), and by use of them, we prove that, for tn→ +∞, the solution of (S) converges to some solution of the elliptic system associated with (S).

f = A áH and g = pH, we prove that the solution of (5) converges to a solution of the Dirichlet problem for the elliptic system.
We obtain regularizing effects such that : Our method is closely related to the paper of LANGLAIS and PHILLIPS [7], and also to the paper of ELHACHIMI and DE THELIN [3] who study the stabilization of the solution of a single equation .Some examples are discussed in part IV, and include : Some numerical results related to the system (S) are given in [4] .All Theoem are written in the case p > 2, q > 2; obvious modifications (for Theorem 6) give the case p = 2 or q = 2. Throughout this paper, 52 stands for a regular bounded open subset of RN .Left f and g be some functions from R N+2 t o R such that: (1.1) f, 9EC1(2xRxR) and for any x E 52, u E R+ , V E R+ : f (x, 0, v) > 0, g(x, u, 0) > 0 and át (IVUIP-2 Vu) E L2(to,+oo ; LP*( 52)) and at (lovl9-2 w) E L2(to,-+-oo ; L9*(S2)) .

. Preliminaries and sub-supersolutions
For any M > 0, N > 0, there exist kM,N > 0, k2 M N > 0 such that a Remark 1 : the condition 1 .2. a) is satisfied if u + f (x, u, v) is a non increasing function on R+ .
Let cp o , Oo be given such that : We say that (u, v) is a solution of (S) in QT (resp: (ú, v) is a supersolution of (S) in QT) iff Our method is based upon a comparison principale for the system (S); but the usual notion of supersolution does not work; so, following Hernandez [6], we set : Definition 1. [(0, 0), (ú, v)] is said to be a sub-supersolution of (S) in QT if it satisfies the following conditions : Remark 2: if we suppose that v -> f (x, u, v) 1 and u + g(x, u, v) 1 any supersolution of (S) gives a sub-supersolution of (S) .
Our first results are sufficient conditions for the existence of sub-supersolutions of (S) .
Theorem 2 .Le¡ v -> f(x, u, v) be a non-decreasing function and u g(x, u, v) be a non-increasing function .
Assume that there exitt constants and for any N: such that : Then (S) has a sub-supersolution.
Proof.By [3] there exits v such that : -O q v > /to + /L1 v7z Let N < v and u be such that : -O pio < Ao + Al u-r1 Then: whence the result .
Proof of Lemma 2: By lemma 1, for any n E N, u n and vn are bounded ; whence (2.8) .The properties of the functions f and g, then imply that f (X, un+l, vn) is bounded .
We therefore obtain: It is the same for vn+l .
Proof of theorem 4: By (2.8), (2.9), (2.10), there is a subsequence (un, vn) with the following properties: Un converges to u in the weak * sense in L°°(0, T ; Wol'p(Q) fl Lw(Q)) and un converges weakly in LP(0, T ; W, 1 `(Q» ; un is such that aát converges to áa in weak LZ(QT) ; the same holds also for vn with p replaced by q .
Remark 3: Uniqueness follows from the Lipschitz condition on f and g.
w(90, 00) = {w Our main result is the following :
The lame estimates hold for v provided p and F are replaced by q and G respectively .