Attractors for a Class of Doubly Nonlinear Parabolic Systems ∗

In this paper, we establish the existence and boundedness of solutions of a doubly nonlinear parabolic system. We also obtain the existence of a global attractor and the regularity property for this attractor in [L 1 ()] 2 and 2 Y i=1 B 1+�i,pi 1 ().


Introduction
This paper deals with the doubly nonlinear parabolic system of the form (S) on Ω, on Ω.
Where Ω is a bounded and open subset in R N , (N ≥ 1) with a smooth boundary ∂Ω, T > 0. The operator ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian.
Monotone operators, in particular the ones that are subdifferentials of convex functions, like p-Laplacian, appear frequently in equations modeling the behaviour of viscoelastic materials (see [16] for instance), reaction-diffusion (see [17], and references therein) and in mathematical glaciology.
Here, we study the existence of solutions for a class of doubly nonlinear systems including the p-Laplacian as the principal part of the operator, and we use the general setting of attractors ( see [19]) to prove that all the solutions converge to a set A, which is called the global attractor.In fact, few papers consider the question in such situations.For instance, Marion [17] considered the problem of solutions of reaction-diffusion systems in which b i (s) = s and p 1 = p 2 = 2. L.Dung [13,14] treated a system involving the p-Laplacian and b i (s) = s, and proved that weak L q dissipativity implies strong L ∞ one for solutions of degenerate nonlinear diffusion systems and gives the existence of global attractors to which all solutions converge in uniform norm.We mention that to our knowledge, the doubly nonlinear parabolic system for the p-Laplacian operator has never been studied, not even in the case b i (s) = s.In the classical setting, i.e with p 1 = p 2 = 2, the system with b i has been previously considered, for example in [9] and [10].We follow the approach of [10], generalizing some results to the case p i > 1 and we extend the results of [11] to nonlinear system (S).In the first section of this paper, we give some assumptions and preliminaries, in section 2 and section 3, we prove the existence of an absorbing set and the existence of the attractor, in section 4, we present the regularity of the attractor and obtain the asymptotic behaviour of the solutions in the framework of dynamical systems associated to the system (S).
2 Preliminaries, Existence and Uniqueness

Notations and Assumptions
We shall assume throughout the paper that Ω is a regular open bounded subset of R N and for any T > 0, we set Q T = Ω × (0, T ) and S T = ∂Ω × (0, T ), with ∂Ω 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 ≤ +∞ . .1,q if X = W 1,q (Ω) for all q : 1≤ q ≤ +∞ , .X otherwise and ., .X,X will denote the duality product between X and its dual X .We use the standard notation for Sobolev spaces W 1,r 0 (Ω), 1 < r < +∞, and their duals W −1,r (Ω), where r = r/(r − 1).The following lemma are useful and frequently used : Lemma 2.1 ( Ghidaghia lemma, cf [19]) Let y be a positive absolutely continuous function on (0,∞) which satisfies y + µy q ≤ λ, Lemma 2.2 ( Uniform Gronwall's lemma, cf [19]) Let y and h be locally integrable functions such that : We start by introducing our assumptions and making precise the meaning of solution of (S).
We shall assume that the following hypotheses are satisfied : (H2) b i ∈ C 1 (R), b i (0) = 0, and there exist positive constants γ i and M i such that (H4) a) There exists positive constants c 1 > 0, c 2 > 0, c 3 > 0 and α 1 > sup(2, p 1 ) such that for any ξ ∈ R any N > 0 we have for any u 2 : where a 1 : R + → R + is an increasing function.
(H5) ∂fi ∂t (x, t, η, ζ) exist and for all L > 0, there exists (H6) a) There exist δ 1 > 0 such that for almost every (x, t) ∈ Ω × R + and for any N > 0 and any u 2 : b) There exist δ 2 > 0 such that for almost every (x, t) ∈ Ω × R + and for any M > 0 and any u 1 : Definition 2.1 By a weak solution of (S), we mean an element w = (u 1 , u 2 ) : and f or all where

Existence
We have the following.
Theorem 2.1 Let the general assumptions (H1)-(H7) be satisfied, then for any τ > 0, the problem (S) has a weak solution (u 1 , u 2 ) such that Proof.By the existence of theorem [11, theorem 3.1, p.3] , there exists two functions u 0 1 , u 0 2 solutions of the problem 2 ) we construct two sequences of functions (u n 1 ) , (u n 2 ) such that And The existence of solutions will be shown via some a-priori L ∞ estimates on (u n 1 , u n 2 ) and lemma 2.3 and lemma 2.4.In all this paper, we denote by c i different constants independent of n and depending on p i , Ω, T. Sometimes we shall refer to a constant depending on specific parameters : c(τ ), c(T ), c(τ, T ).
Lemma 2.3 Under the hypothesis (H1)-(H7), there exist c i > 0 such that for any n ∈ N and any τ > 0, the following estimate holds Proof.The case n = 0 has been observed.Assume that (2.7) is valid for (n−1) and let us derive the estimate for n.Now multiplying (2.1) by |b 1 ( u n 1 )| k b 1 ( u 1 ) and using the growth condition on b 1 , and (H4) a) we deduce for all τ > 0 Setting y k,n (t) = b 1 ( u n 1 ) L k+2 (Ω) and using Hölder inequality on both sides, there exists two constants λ 0 > 0 and µ 0 > 0 such that which implies from a lemma 2.1 that As k → +∞, and for any all t ≥ τ > 0, we have by (2.10) and (H2) The same holds for u n (2.12) Lemma 2.4 Under the hypothesis (H1)-(H7), for all τ > 0, there exists constants c j , c τ such that the following estimates hold EJQTDE, 2006 No. 1, p. 5 ) Proof.Taking the scalar product of equation (2.1) by u n 1 and (2.4) by u n 2 , integrating on Ω and using hypothesis (H4), we get (2.17) So, integrating (2.17) from 0 to T we obtain ∂t integrating on Ω, it follows by (H2),(H7) and lemma 2.1 that for any all t ≥ τ > 0, Then, we have Integrating (2.20) on (t, t + τ ) , then yields Integrating (2.17) on (t, t + τ ) and using lemma 2.3, we get By the uniform Gronwall's lemma 2.1, we obtain Integrating now (2.20) on (t, t + τ ) , we have which gives by (H2) Passage to the limit in in the process (P 1,n ) and (P 2,n ) By lemma 2.3 and lemma 2.4, there exist a subsequence (denoted again by u n i , i = 1, 2) such that as n → +∞: is bounded in L 2 (τ, T ; W −1,p i (Ω)) for any τ > 0, divF i (∇u n i ) → χ i in weak L p i (0, T ; W −1,p i (Ω)).Moreover standard monotonicity argument gives χ i = divF i (∇u), η i = b i (u i ).To conclude that w = (u 1 , u 2 ) is a weak solution of (S) it is enough to observe that f 1 (x, t, u n 1 , u n−1
Remark.i) Our calculations above are formal.We may assume that the solutions are smooth enough to have all estimates we need.Such assumptions EJQTDE, 2006 No. 1, p. 8 can be justified by working with regularized problem whose solutions are smooth so that the following argument can be carried out rigorously.One can see that the estimates obtained are independent of the parameter ε, so that, by taking the limit, they also hold for (S).
ii) Assume that hypothesis (H1) to (H7) are satisfied and f i does not depend on t : , then theorem 2.1 establishes the existence of dynamical system {S(t)} t≥0 which maps L 2 (Ω) 3 Global attractor Proposition 3.1 Assume that (H1)-(H7) hold and f i does not depend on t, the semi-group S(t) associated with problem (S) is such that (i) There exist absorbing sets in L σi (Ω), for 1 ≤ σ i ≤ +∞.
(ii) There exist absorbing sets in W 1,p1 0 Proof.Let u i be solution of (S) and u n i solution of (P i,n ) such that u n i → u i .Then for fixed t ≥ τ > 0, lemma 2.3, lemma 2.4 and Sobolev's injection theorem imply As n → +∞, we get Remark.By proposition 3.1 we deduce that the assumptions (1.1),(1.4)and (1.12) in theorem 1.1 [19] p23 are satisfied with U = L 2 (Ω) 2 .So, by means of the uniform compactness lemma in [7, p. 111], we get the following result.
Theorem 3.1 Assume that (H1)-(H7) are satisfied and that f i does not depend on time.Then the semi-group S(t) associated with the boundary value problem Its domain of attraction is the whole space L 2 (Ω) 2 .
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 there exist positive constants δ i > 0 and a function H from R N +2 to R such that : (H8) H4),(H5) and (H6), The following lemmas are used in the proof of the main results of this section.
which yields So that, the Hölder inequality can be applied to give then yields (4.5).For stating the next theorem we introduce the hypothesis (H10) N = 1 and 1 < p i < 2 or N ≥ 2 and 3N N +2 ≤ p i < 2. where c 25 is a positive constant depending on τ.