We consider the

We prove existence and asymptotic behaviour results for weak solutions of a mixed problem (S). We also obtain the existence of the global at- tractor and the regularity for this attractor in H 2 () 2 and we derive estimates u1 = u2 = 0 in @ (0;T ) (b1(u1(x; 0);b2(u2(x; 0)) = (b1('0(x));b2( 0(x))) in where is a bounded open subset in R N , N 1, with a smooth boundary @ : (S) is an example of nonlinear parabolic systems modelling a reaction dif- fusion process for which many results on existence, uniqueness and regularity have been obtained in the case where bi(s) = s ( see, for instance (6; 7; 18)). The case of a single equation of the type (S) is studied in (1; 2; 3; 4; 5; 8; 9; 19): The purpose of this paper is the natural extension to system (S) of the results by (8), which concerns the single equation @ (u) @t u +f(x;t;u) = 0: Actually, our work generalizes the question of existence and regularity of the global attractor obtained therein. In the rst section of this paper, we give some assumptions and preliminaries and in section 2, we prove the existence of absorbing sets and the existence of the gobal attractor; while in section 3, we present the regularity of the attractor and show stabilization property. Finally, section 4 is devoted to estimates of the Haussdorf and fractal dimensions. 1. Preliminaries, Existence and Uniqueness 1.1 Notations and Assumptions Letbi, (i = 1; 2) be continuous functions withbi(0) = 0: We dene fort 2 R i(t) = R t 0 bi( )d : Then the Legendre transform of is dened by i ( ) = sup2R f s i(s)g: stands for a regular open bounded subset of


Introduction
We consider the following nonlinear system (S ) where Ω is a bounded open subset in R N , N ≥ 1, with a smooth boundary ∂Ω.(S) is an example of nonlinear parabolic systems modelling a reaction diffusion process for which many results on existence, uniqueness and regularity have been obtained in the case where b i (s) = s ( see, for instance [6,7,18]).
The case of a single equation of the type (S) is studied in [1,2,3,4,5,8,9,19] .The purpose of this paper is the natural extension to system (S) of the results by [8], which concerns the single equation ∂β(u)  ∂t − ∆u + f (x, t, u) = 0. Actually, our work generalizes the question of existence and regularity of the global attractor obtained therein.
In the first section of this paper, we give some assumptions and preliminaries and in section 2, we prove the existence of absorbing sets and the existence of the gobal attractor; while in section 3, we present the regularity of the attractor and show stabilization property.Finally, section 4 is devoted to estimates of the Haussdorf and fractal dimensions.

Preliminaries, Existence and Uniqueness
1.1 Notations and Assumptions Let b i , (i = 1, 2) be continuous functions with b i (0) = 0. We define for t ∈ R Ψ i (t) = R N and for any T > 0, we set Q T = Ω × (0, T ) and S T = ∂Ω × (0, T ), where ∂Ω is the boundary of Ω .The norm in a space X will be denoted by : .r if X = L r (Ω) for all r : 1≤ r ≤ +∞ , .X otherwise and ., .X,X will denote the duality product between X and its dual X .
We start by introducing our assumptions and making precise the meaning of a solution of (S).Consider the system ( S) under the following assumptions: , and there exists c ij > 0 such that : Definition By a weak solution of (S), we mean an element where Theorem 1 Let (H1) to ( H6) be satisfied.Then there exists a solution (u 1 , u 2 ) of problem (S) such that for i = 1, 2 , we have in Ω and we construct two sequences of functions (u n 1 ) and (u n 2 ), such that : in Ω (1.6)We need lemma 1 and lemma 2 below to complete the proof of theorem 1.
From lemma 2 and Lemma 1, there is a subsequence u n i (i = 1, 2) with the following properties: ( by the compactness result of Aubin ( see [22])).By lemma 7([9]), we have ), ∀r ≥ 1; taking the limit as n goes to ∞ , we deduce that ( u 1 , u 2 ) is a weak solution of (S).

A regularity property of the attractor
In this section we shall show supplementary regularity estimates on the solution of problem (S) and by use of them, we shall obtain more regularity on the attractor obtained in section 3. We shall assume that Hereafter, we shall assume that there exist positive constants δ i > 0 and a function Φ from R N +2 to R such that : f i satisfying (H3) to (H6) and h i ∈ L ∞ (Ω).
For a solution (u 1 , u 2 ) of (S), we define the ω − limit set by : .
Corollary 2. Under the assumptions (H1) to (H10), we have Proof: Taking the inner product of (4.7) with u i , we get ).
By uniform Gronwall's lemma, we get Then

Linearized problem
Let (ϕ 0 , ψ 0 ) ∈ A; then by theorem3, u(t) = (u 1 (t), u 2 (t)) belongs to a bounded subset of H 2 (Ω) 2 .This fact allows us to linearize the system ( S) along u(t).Formally, the candidate for the linearized problem is The existence and uniqueness of solution can be deduced from (4.0) below To deduce (4.0), Multiply the equation in (S L ) by b i (u i )U i , we obtain By the hypothesis on b i , we have ( From (4.2) to (4.4), (4.1) becomes 1 2 By standard application of Gronwall's inequality, we get (4.0).
Proof of proposition 3 : It is similar to the proof for the lemma 15 in [8, p.125] and is omited.