Electronic Journal of Qualitative Theory of Differential Equations 2006, No. 1, 1-15; http://www.math.u-szeged.hu/ejqtde/

Attractors for a Class of Doubly Nonlinear Parabolic Systems^{∗}

Hamid El Ouardi & Abderrahmane El Hachimi

Abstract

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^{∞}(Ω)]^{2}

and ∏_{i=1}^{2}B_{∞}^{1+σ_{i},p_{i}}(Ω).

1 Introduction

This paper deals with the doubly nonlinear parabolic system of the form

(S) \{ \begin{matrix}\begin{matrix}\frac{∂b_{1}(u_{1})}{∂t} − Δ_{p_{1}}u_{1} + f_{1}(x, t, u_{1}, u_{2}) = 0 \\ \frac{∂b_{2}(u_{2})}{∂t} − Δ_{p_{2}}u_{2} + f_{2}(x, t, u_{1}, u_{2}) = 0 \end{matrix} & \begin{matrix}in Ω × (0, ∞), \\ in Ω × (0, ∞), \end{matrix} \\ u_{1} = u_{2} = 0 & in ∂Ω × (0, ∞), \\ \begin{matrix}b_{1}(u_{1}(., 0)) = b_{1}(φ_{1}) \\ b_{2}(u_{2}(., 0)) = b_{2}(ψ_{1}) \end{matrix} & \begin{matrix}on Ω, \\ on Ω. \end{matrix} \end{matrix}

Where Ω is a bounded and open subset in ℝ^{N}, (N ≥ 1) with a smooth boundary ∂Ω, T > 0. The operator Δ_{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).

∗Mathematics Subject Classifications: 35K55, 35K57, 35K65, 35B40, 35B45.

Key words: parabolic systems, p-Laplacian, global attractor, asymptotic behaviour.

2 Preliminaries, Existence and Uniqueness

2.1 Notations and Assumptions

Let b_{i}, (i = 1, 2) be a continuous function with b_{i}(0) = 0. For t ∈ ℝ ,define, Ψ_{i}(t) = ∫_{0}^{t} b_{i}(s)ds. The Legendre transform Ψ_{i}^{∗} of Ψ_{i} is defined as Ψ_{i}^{∗}(τ) = sup_{s∈ℝ}{τs − Ψ_{i}(s)}. 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_{0}^{1,r}(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,

with q > 1, μ > 0, λ ≥ 0. Then for t > 0

y(t) ≤ (\frac{λ}{μ})^{\frac{1}{q}} + [μ(q − 1)t]^{\frac{−1}{q−1}} .

Lemma 2.2 ( Uniform Gronwall’s lemma, cf [19]) Let y and h be locally integrable functions such that :

∃r > 0, a_{1} > 0, a_{2} > 0, τ > 0, ∀t ≥ τ

∫_{t}^{t+r} y(s)ds ≤ a_{1}, ∫_{t}^{t+r} |h(s)| ds ≤ a_{2}, y′ ≤ h.

Then

y(t + r) ≤ \frac{a_{1}}{r} + a_{2}, ∀t ≥ τ.

We start by introducing our assumptions and making precise the meaning of solution of (S).

We shall assume that the following hypotheses are satisfied :

(H1) (φ_{1}, ψ_{1}) ∈ [L^{2}(Ω)]^{2} .

(H2) b_{i} ∈ C^{1}(ℝ), b_{i}(0) = 0, and there exist positive constants γ_{i} and M_{i} such

that

|b_{i}| ≤ γ_{i} |s| + M_{i}, ∀s ∈ ℝ, i = 1, 2.

(H3) f_{i} ∈ C^{1}(Ω × ℝ × ℝ).

(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} : |u_{2}| < N

sign(ξ)f_{1}(x, t, ξ, u_{2}) ≥ c_{1} |b_{1}(ξ)|^{α_{1}−1} − c_{2} ,

lim_{t→0^{+}} sup |f_{1}(x, t, ξ, u_{2})| ≤ c_{3} (|ξ|^{α_{1}−1} + 1)

|f_{1}(x, t, ξ, u_{2})| ≤ a_{1}(|ξ|) almost everywhere in Ω × ℝ^{+} where a_{1} : ℝ^{+} → ℝ^{+} is an increasing function.

b) There exists positive constants c_{4} > 0, c_{5} > 0, c_{6} > 0 and α_{2} > sup(2, p_{2}) such that for any ξ ∈ ℝ any M > 0 we have for any u_{1} : |u_{1}| < M

sign(ξ)f_{2}(x, t, u_{1}, ξ) ≥ c_{4} |b_{2}(ξ)|^{α_{2}−1} − c_{5} ,

lim_{t→0^{+}} sup |f_{2}(x, t, u_{1}, ξ)| ≤ c_{6} (|ξ|^{α_{2}−1} + 1)

|f_{2}(x, t, u_{1}, ξ)| ≤ a_{2}(|ξ|) almost everywhere in Ω × ℝ^{+}

where a_{2} : ℝ^{+} → ℝ^{+} is an increasing function.

(H5) \frac{∂f_{i}}{∂t} (x, t, η, ζ) exist and for all L > 0, there exists C_{L} > 0 such that : if

|η| + |ζ| ≤ L then |\frac{∂f_{i}}{∂t} (x, t, η, ζ)| ≤ C_{L}, for almost every (x, t) ∈ Ω × ℝ^{+}.

(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} : |u_{2}| < N then

ξ → f_{1}(x, t, ξ, u_{2}) + δ_{1}b_{1}(ξ) is increasing.

b) There exist δ_{2} > 0 such that for almost every (x, t) ∈ Ω × R^{+} and for any M > 0 and any u_{1} : |u_{1}| < M then

ξ → f_{2}(x, t, u_{1}, ξ) + δ_{2}b_{2}(ξ) is increasing.

(H7) ∃ε > 0 : b′_{i}(s) ≥ ε, for all s ∈ ℝ.

Definition 2.1 By a weak solution of (S), we mean an element w = (u_{1}, u_{2}) :

u_{i} ∈ L^{p_{i}}(0, T; W_{0}^{1,p_{i}}(Ω)) ∩ L^{α_{i}}(Q_{T}) ∩ L^{∞}(τ, T; L^{∞}(Ω)) for all τ > 0,

\frac{∂b_{i}(u_{i})}{∂t} ∈ L^{p′_{i}}(0, T; W^{−1,p′_{i}}(Ω)) + L^{α′_{i}}(Q_{T}),

and for all ϕ_{i} ∈ L^{p_{i}}(0, T; W_{0}^{1,p_{i}}(Ω)) ∩ L^{∞}(0, T; L^{∞}(Ω))

∫_{0}^{T} ⟨\frac{∂b_{i}(u_{i})}{∂t}, ϕ_{i}⟩_{X_{i},X′_{i}} dt + ∫_{0}^{T} ∫_{Ω} F_{i}(∇u_{i})∇ϕ_{i}dxdt = − ∫_{0}^{T} ∫_{Ω} f_{i}(x, w)ϕ_{i}dxdt

and if \frac{∂ϕ_{i}}{∂t} ∈ L^{2}(0, T; L^{2} (Ω)), ϕ_{i}(T) = 0 then

∫_{0}^{T} ⟨\frac{∂b_{i}(u_{i})}{∂t}, ϕ_{i}⟩_{X_{i},X′_{i}} dt = − ∫_{0}^{T} ∫_{Ω} (b_{i}(u_{i}) − b_{i}(u_{i}(., 0))) \frac{∂ϕ_{i}}{∂t} dxdt,

where X_{i} = L^{∞}(Ω) ∩ W_{0}^{1,p_{i}}(Ω), X′_{i} = L^{1}(Ω) + W^{−1,p′_{i}}(Ω) and F_{i}(ξ) = |ξ|^{p_{i}−2} ξ for any ξ ∈ ℝ^{N}.

2.2 Existence

2.2.1 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

u_{i} ∈ L^{p_{i}}(0, T; W_{0}^{1,p_{i}}(Ω)) ∩ L^{∞}(τ, T; W_{0}^{1,p_{i}}(Ω) ∩ L^{∞}(Ω)),
and b_{i}(u_{i}) ∈ L^{α_{i}}(Q_{T}) ∩ L^{∞}(0, T; L^{2}(Ω)).

Proof. By the existence of theorem [11, theorem 3.1, p.3] , there exists two functions u_{1}^{0}, u_{2}^{0} solutions of the problem

(P_{1,0}) \{ \begin{matrix}\frac{∂b_{1}(u_{1}^{0})}{∂t} − Δ_{p_{1}}u_{1}^{0} + f_{1}(x, t, u_{1}^{0}, 0) = 0  in Q_{T} \\ u_{1}^{0} = 0  on S_{T} \\ b_{1}(u_{1}^{0}(., 0)) = b_{1}(φ_{1})  in Ω \end{matrix}

(P_{2,0}) \{ \begin{matrix}\frac{∂b_{2}(u_{2}^{0})}{∂t} − Δ_{p_{2}}u_{2}^{0} + f_{2}(x, t, 0, u_{2}^{0}) = 0  in Q_{T} \\ u_{2}^{0} = 0  on S_{T} \\ b_{2}(u_{2}^{0}(., 0)) = b_{2}(ψ_{1})  in Ω \end{matrix}

and u_{i}^{0} ∈ L^{p_{i}}(0, T; W_{0}^{1,p_{i}}(Ω)) ∩ L^{∞}(τ, T; W_{0}^{1,p_{i}}(Ω) ∩ L^{∞}(Ω)) , ∀τ > 0. By (u_{1}^{0}, u_{2}^{0}) we construct two sequences of functions (u_{1}^{n}) , (u_{2}^{n}) such that

(P_{1,n}) \{ \begin{matrix}\frac{∂b_{1}(u_{1}^{n})}{∂t} − Δ_{p_{1}}u_{1}^{n} + f_{1}(x, t, u_{1}^{n}, u_{2}^{n−1}) = 0 & in Q_{T} , \\ u_{1}^{n} = 0 & in S_{T} , \\ b_{1}(u_{1}^{n}(., 0)) = b_{1}(φ_{1}) & on Ω. \end{matrix} \begin{matrix}(2.1) \\ (2.2) \\ (2.3) \end{matrix}

And

(P_{2,n}) \{ \begin{matrix}\frac{∂b_{2}(u_{2}^{n})}{∂t} − Δ_{p_{2}}u_{2}^{n} + f_{2}(x, t, u_{1}^{n−1}, u_{2}^{n}) = 0 & in Q_{T} , \\ u_{2}^{n} = 0 & in S_{T} , \\ b_{2}(u_{2}^{n}(., 0)) = b_{2}(ψ_{1}) & on Ω. \end{matrix} \begin{matrix}(2.4) \\ (2.5) \\ (2.6) \end{matrix}

The existence of solutions will be shown via some a-priori L^{∞} estimates on (u_{1}^{n}, u_{2}^{n}) 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

∥u_{i}^{n}∥_{L^{∞}(τ,T;L^{∞}(Ω))} ≤ c_{7}(τ, T).  (2.7)

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_{1}^{n})|^{k} b_{1}( u_{1}) and using the growth condition on b_{1}, and (H4) a) we deduce for all τ > 0

\frac{1}{k + 2} \frac{d}{dt} \{∫_{Ω} |b_{1}( u_{1}^{n})|^{k+2} dx\} + c_{8} ∫_{Ω} |b_{1}(u_{1}^{n})|^{k+α_{1}} dx ≤

c_{9} ∫_{Ω} |b_{1}(u_{1}^{n})|^{k+1} dx.  (2.8)

Setting y_{k,n}(t) = ∥b_{1}( u_{1}^{n})∥_{L^{k+2}(Ω)} and using Hölder inequality on both sides, there exists two constants λ_{0} > 0 and μ_{0} > 0 such that

\frac{dy_{k,n}(t)}{dt} + μ_{0}y_{k,n}^{α_{1}−1}(t) ≤ λ_{0},  (2.9)

which implies from a lemma 2.1 that

y_{k}(t) ≤ (\frac{λ_{0}}{μ_{0}} )^{\frac{1}{q_{1}−1}} + \frac{1}{[μ_{0}(α_{1} − 2)t]^{\frac{1}{α_{1}−2}}} = c_{10}(t) ∀ t > 0.  (2.10)

As k → +∞, and for any all t ≥ τ > 0, we have by (2.10) and (H2)

∥ u_{1}^{n}(t)∥_{L^{∞}(Ω)} ≤ c_{11}(τ).  (2.11)

The same holds for u_{2}^{n}

∥ u_{2}^{n}(t)∥_{L^{∞}(Ω)} ≤ c_{12}(τ).  (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

∥u_{i}^{n}∥_{L^{p_{i}} (0,T;W_{0}^{1,p_{i}} (Ω))} ≤ c_{13}(T),  (2.13)

∥u_{i}^{n}∥_{L^{∞}(τ,T;W_{0}^{1,p_{i}} (Ω))} ≤ c_{14}(τ, T),  (2.14)
∫_{τ}^{T} ∫_{Ω} b′_{i}( u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dxds ≤ c_{15}(τ, T),  (2.15)

and ∫_{t}^{t+τ} ∫_{Ω} b′_{i}(u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dxds ≤ c_{16}(τ), for any t ≥ τ > 0.  (2.16)

Proof. Taking the scalar product of equation (2.1) by u_{1}^{n} and (2.4) by u_{2}^{n}, integrating on Ω and using hypothesis (H4), we get

\frac{d}{dt} [∑_{i=1}^{2} \{∫_{Ω} Ψ_{i}^{∗}(b_{i}(u_{i}^{n}))dx\}] + ∑_{i=1}^{2} ∫_{Ω} |∇u_{i}^{n}|^{p_{i}} dx

+c_{1} ∑_{i=1}^{2} ∫_{Ω} | u_{i}^{n}|^{α_{i}} dx ≤ c_{2}.  (2.17)

But ∥φ_{1}∥_{L^{2}(Ω)}+∥ψ_{1}∥_{L^{2}(Ω)} ≤ c ⟹ ∫_{Ω} (Ψ_{1}^{∗} ( b_{1}(φ_{1})) + Ψ_{2}^{∗} ( b_{2}( ψ_{1}))) dx ≤ c, where

Ψ_{i}^{∗} is the Legendre transform of Ψ_{i}, Ψ_{i}(t) = ∫_{0}^{t} b_{i}(s)ds. So, integrating (2.17) from 0 to T we obtain

∑_{i=1}^{2} (∫_{0}^{T} ∫_{Ω} |∇u_{i}^{n}|^{p_{i}}) dxds + c_{17}∑_{i=1}^{2} (∫_{0}^{T} ∫_{Ω} | u_{i}^{n}|^{α_{i}}) dxds ≤ c_{17}(T). (2.18)

Hence (2.13) follows. Taking the scalar product of equation (2.1) by \frac{∂u_{1}^{n}}{∂t} and (2.4) by\frac{∂u_{2}^{n}}{∂t} integrating on Ω, it follows by (H2),(H7) and lemma 2.1 that for any all t ≥ τ > 0,

∑_{i=1}^{2} ∫_{Ω} b′_{i}(u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dx + \frac{d}{dt} ∑_{i=1}^{2} [\frac{1}{p_{i}} ∫_{Ω} |∇u_{i}^{n}|^{p_{i}} dx] =

− ∫_{Ω} f_{1}(x, t, u_{1}^{n}, u_{2}^{n−1})\frac{∂u_{1}^{n}}{∂t} dx − ∫_{Ω} f_{2}(x, t, u_{1}^{n−1}, u_{2}^{n})\frac{∂u_{2}^{n}}{∂t} dx

≤ \frac{1}{2} ∑_{i=1}^{2} ∫_{Ω} b′_{i}(u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dx + c_{18}(τ).  (2.19)

Then, we have

∑_{i=1}^{2} ∫_{Ω} b′_{i}(u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dx + \frac{d}{dt} ∑_{i=1}^{2} [\frac{2}{p_{i}} ∫_{Ω} |∇u_{i}^{n}|^{p_{i}} dx] ≤ c_{19}(τ).  (2.20)

Integrating (2.20) on (t, t + τ) , then yields

∑_{i=1}^{2} ∫_{t}^{t+τ} ∫_{Ω} b′_{i}(u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dx + ∑_{i=1}^{2} [\frac{2}{p_{i}} ∫_{Ω} |∇u_{i}^{n}(t + τ)|^{p_{i}} dx] =

∑_{i=1}^{2} [\frac{2}{p_{i}} ∫_{Ω} |∇u_{i}^{n}(τ)|^{p_{i}} dx] + c_{τ} .  (2.21)

Integrating (2.17) on (t, t + τ) and using lemma 2.3, we get

∑_{i=1}^{2} ∫_{t}^{t+τ} [\frac{1}{p_{i}} ∫_{Ω} |∇u_{i}^{n}(s)|^{p_{i}} dxds] ≤ c_{τ} , ∀t ≥ τ > 0.  (2.22)

By the uniform Gronwall’s lemma 2.1, we obtain

∑_{i=1}^{2} [∫_{Ω} |∇u_{i}^{n}(t)|^{p_{i}} dx] ≤ c_{τ} , ∀t ≥ τ > 0, ∀n ∈ ℕ^{∗}.

Integrating now (2.20) on (t, t + τ) , we have

∑_{i=1}^{2} ∫_{t}^{t+τ} ∫_{Ω} b′_{i}(u_{i}^{n})(\frac{∂u_{i}^{n}}{∂t})^{2}dxds ≤ c_{20}(τ),

which gives by (H2)

∑_{i=1}^{2} ∫_{t}^{t+τ} ∫_{Ω}(\frac{∂b_{i}u_{i}^{n}}{∂t})^{2}dxds ≤ c_{21}(τ).

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_{i}^{n}, i = 1, 2) such that as n → +∞: u_{i}^{n} → u_{i} weak in L^{p_{i}}(0, T; W_{0}^{1,p_{i}}(Ω)) and in L^{α_{i}}(Q_{T}), u_{i}^{n} → u_{i}^{n} weak star in L^{∞}(τ, T; W_{0}^{1,p_{i}}(Ω), ∀ τ > 0, b_{i}(u_{i}^{n}) → η_{i} in L^{2}(Q_{T}), \frac{∂b_{i}(u_{i}^{n})}{∂t} is bounded in L^{2}(τ, T; W^{−1,p′_{i}}(Ω)) for any τ > 0, divF_{i}(∇u_{i}^{n}) → 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_{1}^{n}, u_{2}^{n−1}) converges to f_{1}(x, t, u_{1}, u_{2}) and f_{2}(x, t, u_{1}^{n−1}, u_{2}^{n}) converges to f_{2}(x, t, u_{1}, u_{2}) strongly in L^{1}(Q_{T}) and in L^{s}(τ, T; L^{s}(Ω)) for all τ > 0; and for all s ≥ 1, thanks to Vitali’s theorem. Whence w = (u_{1}, u_{2}) is a solution of (S).

2.2.2 Uniqueness

Proposition 2.1 The solution of (S) is unique. Moreover, if (u_{1}, u_{2}) and (v_{1},v_{2}) are two solutions, corresponding respectively to initial data (φ_{1}, ψ_{1}) and (φ_{2}, ψ_{2}) such that φ_{1} ≤ ψ_{1} and φ_{2} ≤ ψ_{2} then u_{i} ≤ v_{i}.

Proof. Suppose that (u_{1}, u_{2}) and (v_{1},v_{2}) are two solutions, corresponding respectively to initial data (φ_{1}, ψ_{1}) and (φ_{2}, ψ_{2}) such that φ_{1} ≤ ψ_{1} and φ_{2} ≤ ψ_{2}. Following Diaz [5, p.269], we consider the following test function : w_{i} = H_{n}(u_{i}− v_{i}), n ≥ 1, (i = 1, 2) by

H_{n}(s) = \{ \begin{matrix}0  if s ≤ 0, \\ \frac{n^{2}s^{2}}{2}  0 < s ≤ \frac{1}{n}, \\ 2ns − \frac{n^{2}s^{2}}{2} − 1  \frac{1}{n} < s ≤ \frac{2}{n}, \\ 1  s > \frac{2}{n}. \end{matrix}

It is easy to see that

0 ≤ (H_{n})′(s) ≤ n, lim_{n→+∞} s(H_{n})′(s) = 0.

|H_{n}(s)| ≤ 1, lim_{n→+∞}H_{n}(s) = sign_{+}(s)

and lim_{n→+∞}s(H_{n})(s) = s_{+} = \{ \begin{matrix}0 s ≤ 0 \\ s s > 0 \end{matrix}

Considering the systems (S) verified by u = (u_{1}, u_{2}) and v=(v_{1},v_{2}) , we get

∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [b_{i}(u_{i}) − b_{i}(v_{i})]_{t} H_{n}(u_{i} − v_{i}) +

∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [|∇u_{i}|^{p_{i}−2} ∇u_{i} − |∇v_{i}|^{p_{i}−2} ∇v_{i}] (∇u_{i} − ∇v_{i})(H_{n})′(u_{i} − v_{i})+

∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [f_{i}(x, u_{1}, u_{2}) − f_{i}(x, v_{1}, v_{2})] H_{n}(u_{i} − v_{i}).  (2.23)

Since (H_{n})′(s) ≥ 0, we deduce that

lim_{n→+∞} ∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [|∇u_{i}|^{p_{i}−2} ∇u_{i} − |∇v_{i}|^{p_{i}−2} ∇v_{i}] (∇u_{i}−∇v_{i})(H_{n})′(u_{i}−v_{i}) ≥ 0.

(2.24)

By (H7) and (2.24), (2.23) becomes

∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [b_{i}(u_{i}) − b_{i}(v_{i})]_{t} sign_{+}(u_{i}−v_{i}) ≤ k_{1} ∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [b_{i}(u_{i}(., s)) − b_{i}(v_{i}(., s)]_{+} ,

(2.25)

by Gronwall’s lemma, we get

∑_{i=1}^{2} ∫_{0}^{t} ∫_{Ω} [b_{i}(u_{i}(., t)) − b_{i}(v_{i}(., t))]_{+} ≤ e^{k_{1}t} ∑_{i=1}^{2} ∫_{Ω} [b_{i}(φ_{i}) − b_{i}(ψ_{i})]_{+} , ∀ t ∈ [0, T] .

(2.26)

Since the second term vanishes and recalling that φ_{1} ≤ ψ_{1} and φ_{2} ≤ ψ_{2}, this means that b_{i}(u_{i}) ≤ b_{i}(v_{i}), and by monotonicity of b_{i}, we obtain u_{i} ≤ v_{i}. Uniqueness is now an obvious consequence.

Remark. i) Our calculations above are formal. We may assume that the solutions are smooth enough to have all estimates we need. Such assumptions can be justified by working with regularized problem

\frac{∂b_{i}(u_{i})}{∂t} − div [\{|∇u_{i}|^{2} + ε\}^{\frac{p_{i}−2}{2}} ∇u_{i}] + f_{i}(x, u) = 0

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 : f_{i}(x, t, u_{1}, u_{2}) = f_{i}(x, u_{1}, u_{2}), then theorem 2.1 establishes the existence of dynamical system {S(t)}_{t≥0} which maps [L^{2}(Ω)]^{2}into [L^{2}(Ω)]^{2} such that S(t)(φ_{1}, ψ_{2}) = (u_{1}(t), u_{2}(t)).

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^{σ_{i}}(Ω), for 1 ≤ σ_{i} ≤ +∞.

(ii) There exist absorbing sets in W_{0}^{1,p_{1}}(Ω) × W_{0}^{1,p_{2}}(Ω).

Proof. Let u_{i} be solution of (S) and u_{i}^{n} solution of (P_{i,n}) such that u_{i}^{n} → u_{i}. Then for fixed t ≥ τ > 0, lemma 2.3, lemma 2.4 and Sobolev’s injection theorem imply

∥u_{i}^{n}(t)∥_{L^{σ_{i}} (Ω)} ≤ c_{τ} ,
and ∥u_{i}^{n}(t)∥_{W_{0}^{1,p_{i}} (Ω))} ≤ c_{τ}, ∀ t ≥ τ.

As n → +∞, we get

∥u_{i}(t)∥_{L^{∞}(Ω)} ≤ c_{τ} ,
and ∥u_{i}(t)∥_{W_{0}^{1,p_{i}} (Ω))} ≤ c_{τ}, ∀ t ≥ τ.

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 (S) possesses a maximal attractor A which is bounded in [W_{0}^{1,p_{1}}(Ω) × W_{0}^{1,p_{2}}(Ω)]∩

[L^{∞}(Ω)]^{2} , compact and connected in [L^{2}(Ω)]^{2} . Its domain of attraction is the whole space [L^{2}(Ω)]^{2} .

4 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 there exist positive constants δ_{i} > 0 and a function H from ℝ^{N+2} to ℝ such that :

(H8) \{ \begin{matrix}f_{i}(x, u) = g_{i}(u) − h_{i}(x) = δ_{i} \frac{∂H}{∂u_{i}}; \\ f_{i} satisfy (H3),(H4),(H5) and (H6), \\ and h_{i} ∈ L^{∞}(Ω) \end{matrix}

(H9) b_{i} ∈ C^{2}(ℝ) and ∃γ_{i}, M_{i} > 0 such that γ_{i} ≤ b′_{i}(s) ≤ M_{i}, ∀s ∈ ℝ. We shall denote : E_{i}(ξ) = |ξ|^{(p_{i}−2)/2}ξ, for all ξ ∈ ℝ^{N}. The following lemmas are used in the proof of the main results of this section.

Lemma 4.1 Assume that (H1)-(H9) are satisfied, there exist constants C = C(φ_{1}, ψ_{1}), such that for any T > 0

∥u_{i}∥_{L^{∞}(0,T,W_{0}^{1,p_{i}} (Ω))} ≤ C < ∞,  (4.1)

and ∥\frac{∂u_{i}}{∂t} ∥_{L^{2}(Q_{T})} ≤ C < ∞.  (4.2)

Proof. Multiplying the equation \frac{∂b_{i}(u_{i})}{∂t} − div [|∇u_{i}|^{p_{i}−2} ∇u_{i}] + δ_{i} \frac{∂H}{∂u_{i}} = 0 by

\frac{1}{δ_{i}} (u_{i})_{t} and we obtain

∑_{i=1}^{2} \frac{1}{δ_{i}} ∫_{Q_{T}} b′_{i}(u_{i})(\frac{∂u_{i}}{∂t})^{2}dxdt + ∑_{i=1}^{2} \frac{1}{p_{i}δ_{i}} ∫_{Ω} |∇u_{i}(., T)|^{p_{i}} dx =  (4.3)

∫_{Ω} [−H(., u_{1}(T), u_{2}(T)) + H(., φ_{1}, ψ_{1}] dx = \frac{1}{p_{1}δ_{1}} ∫_{Ω} |∇φ_{1}|^{p_{1}} dx+\frac{1}{p_{2}δ_{2}} ∫_{Ω} |∇ψ_{1}|^{p_{2}} dx.

H is continuous and (u_{1}, u_{2}) is bounded, we then obtain

∑_{i=1}^{2} \frac{γ_{i}}{δ_{i}} ∫_{Q_{T}} (\frac{∂u_{i}}{∂t})^{2}dxdt + ∑_{i=1}^{2} \frac{1}{p_{i}δ_{i}} ∫_{Ω} |∇u_{i}(., T)|^{p_{i}} dx ≤ C(φ_{1}, ψ_{1}), (4.4)

hence (4.1) and (4.2). ■

Lemma 4.2 Let p_{i} ∈ ]1, 2[, then we have the following estimate

∑_{i=1}^{2} ∫_{Ω} |∇u′_{i}|^{p_{i}} dx ≤ c_{22} ∑_{i=1}^{2} ∫_{Ω} |∇u_{i}|^{p_{i}} dx + ∑_{i=1}^{2} \frac{2(p_{i} − 1)}{p_{i}^{2}} ∫_{Ω} |(E_{i}(∇u_{i}))′|^{2} dx,

(4.5)

with a constant c_{22} > 0.

Proof. Straigthforward calculations see [9] give

∫_{Ω} (F_{i}(∇w_{i}))′ .∇w′_{i}dx ≥ (p_{i} − 1) (\frac{2}{p_{i}} )^{2} ∫_{Ω} |(E_{i}(∇w_{i}))′|^{2} dx.

As E_{i}(∇w_{i}) = |∇w_{i}|^{\frac{p_{i}−2}{2}} ∇w_{i} , we get |∇w_{i}| = |E_{i}(∇w_{i})|^{\frac{2}{p_{i}}} and ∇w_{i} = |E_{i}(∇w_{i}|^{\frac{2−p_{i}}{2}} E(∇w_{i}) Hence

(∇w_{i})′ = \frac{2}{p_{i}} |E_{i}(∇w_{i})|^{\frac{2−p_{i}}{p_{i}}} (E_{i}(∇w_{i}))′ ,

which yields

|∇w′_{i}|^{p_{i}} = (\frac{2}{p_{i}})^{p_{i}} |E_{i}(∇w_{i})|^{2−p_{i}} |(E_{i}(∇w_{i}))′|^{p_{i}} .

So that, the Hölder inequality can be applied to give

∫_{Ω} |∇w′_{i}|^{p_{i}} dx ≤ c_{23} ∫_{Ω} |E_{i}(∇w_{i})|^{2−p_{i}} |(E_{i}(∇w_{i}))′|^{p_{i}} dx

≤ \frac{c_{24}}{2} ∫_{Ω} |E_{i}(∇w_{i})|^{2} dx + \frac{2(p_{i} − 1)}{p_{i}^{2}}(∫_{Ω} |(E_{i}(∇w_{i}))′|^{2} dx,

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 \frac{3N}{N+2} ≤ p_{i} < 2.

Theorem 4.1 Let f_{i}, b_{i} and p_{i} satisfies hypothesis (H1) to (H10).

Let r(t) = ∑_{i=1}^{2} ∫_{Ω} b′_{i}(u_{i}) (ui^{′})^{2} dx. Then

r(t) ≤ c_{25}(τ), ∀ t ≥ τ > 0.  (4.6)

where c_{25} is a positive constant depending on τ.

Proof. Differentiating equation \frac{∂b_{i}(u_{i})}{∂t}−div [|∇u_{i}|^{p_{i}−2} ∇u_{i}]+g_{i}(x, u) = h_{i}(x) with respect to t, we get

b′_{i}(u_{i})u′′_{i} + b′′_{i}(u_{i})(u′_{i})^{2} − div ((F_{i}(∇u_{i}))′) + ∑_{j=1}^{2} \frac{∂g_{i}(u)}{∂u_{j}}u′_{j} = 0.  (4.7)

Now multiplying (4.7) by u′_{i}, and integrating over Ω gives

\frac{1}{2}r′(t)+\frac{1}{2} ∑_{i=1}^{2} ∫_{Ω} b′′_{i}(u_{i})(u′_{i})^{3}dx+∑_{i=1}^{2} ∫_{Ω} (F_{i}(∇u_{i}))′ ∇u′_{i}dx+∫_{Ω} (∑_{i=1}^{2} ∑_{j=1}^{2} \frac{∂g_{i}(u)}{∂u_{j}}u′_{j}) u′_{i}dx = 0,

(4.8) the L^{∞} estimate and hypothesis (H9) imply successively

∫_{Ω} (∑_{i=1}^{2} ∑_{j=1}^{2} \frac{∂g_{i}(u)}{∂u_{j}}u′_{j}) u′_{i}dx ≤ M ∑_{i=1}^{2} ∫_{Ω} (u′_{i})^{2} dx.  (4.9)

γ ∑_{i=1}^{2} ∫_{Ω} (u′_{i})^{2} d ≤ r(t).  (4.10)

On the other hand, by Gagliardo-Nirenberg’s inequality, Young’s inquality and (4.5), we obtain

−\frac{1}{2} ∑_{i=1}^{2} ∫_{Ω} b′′(u_{i})(u′_{i})^{3}dx ≤ c_{25} ∑_{i=1}^{2} ||u′_{i}||_{2}^{3(1+q_{i})} + c_{26} ∑_{i=1}^{2} ||∇u_{i}||_{p_{i}}^{p_{i}} +

∑_{i=1}^{2} \frac{4(p_{i} − 1)}{p_{i}^{2}} ∫_{Ω} |(E_{i}(∇u_{i}))′|^{2} dx,  (4.11)

where q_{i} = \frac{N(3−p_{i})}{3Np_{i}+6p_{i}−9N} . By (4.9),(4.10),(4.11), (4.7) becomes

\frac{1}{2}r′(t) + ∑_{i=1}^{2} \frac{(p_{i} − 1)}{2} (\frac{2}{p_{i}} )^{2} ∫_{Ω} |(E_{i}(∇u_{i}))′|^{2} dx ≤ c_{27} ∑_{i=1}^{2} ||u′_{i}||_{2}^{3(1+q_{i})} +

c_{128} ∑_{i=1}^{2} ||∇u_{i}||_{p_{i}}^{p_{i}} + M ∑_{i=1}^{2} ∥u′_{i}∥_{2}^{2} .  (4.12)

Now (4.11) and estimate (2.13) give

\frac{1}{2}r′(t) + ∑_{i=1}^{2} \frac{2(p_{i} − 1)}{p_{i}^{2}} ∫_{Ω} |(E_{i}(∇u_{i}))′|^{2} dx ≤ c_{29} (r(t))^{2} + c_{30} for any t ≥ τ > 0.

(4.13)

Using estimate (2.14) now gives

∑_{i=1}^{2} [\frac{1}{p_{i}} ∫_{Ω} |∇u_{i}^{n}|^{p_{i}} dx] ≤ c_{31}(τ) ,∀t ≥ \frac{τ}{2} for any τ > 0,

integrating (2.20) on [t, t + \frac{τ}{2} ], then yields

∑_{i=1}^{2} ∫_{t}^{t+ \frac{τ}{2}} ∫_{Ω} b′_{i}(u_{i}) (u′_{i})^{2} dxdt ≤ c_{32}(τ), for any t ≥ \frac{τ}{2} > 0.  (4.14)

That is

∫_{t}^{t+ \frac{τ}{2}} r(s)ds ≤ c_{33}(τ), for any t ≥ \frac{τ}{2} > 0.  (4.15)

Coming back to (4.13) and using the uniform Gronwall’s lemma 2.2 gives

r(t + \frac{τ}{2} ) ≤ c_{34}(τ), for any t ≥ \frac{τ}{2} > 0.

Hence

r(t) ≤ c_{τ} , for any t ≥ τ > 0.

By use of theorem 4.1, we shall now arrive to the aim result of this section.

Theorem 4.2 Let f_{i}, b_{i} and p_{i} satisfies hypothesis (H1) to (H10). Then, for any τ > 0, the solution of system (S) satisfies the following regularity estimates

\frac{∂ b_{i}(u_{i})}{∂t} ∈ L^{2}(τ, +∞; L^{2}(Ω)),  (4.16)
and u_{i} ∈ L^{∞}(τ, +∞; B_{∞}^{1+σ_{i},p_{i}} (Ω)).  (4.17)

Moreover, there exists a constant c_{τ} > 0 such that

lim_{t→+∞} ∥∇u_{i}|^{(p_{i}−2)/2} \frac{∂∇u_{i}}{∂t} ∥_{L^{2}(t,t+1;L^{2}(Ω))} ≤ c(τ).  (4.18)

Proof. By theorem 4.1 and hypothesis (H2), we get :

∑_{i=1}^{2} ∫_{Ω} (\frac{∂b_{i}(u_{i})}{∂t})^{2} dx ≤ Mr(t) ≤ c(τ) for any t ≥ τ > 0, then yields (4.16).

Integrating (4.13) on [t, t + 1], for any t ≥ τ and using theorem 4.1 then yields:

∑_{i=1}^{2} ∫_{t}^{t+1} ∫_{Ω} |(E_{i}(∇u_{i}))′|^{2} dxds ≤ c(τ), for any τ > 0,  (4.19)

whence the estimate (4.18). On the other hand by (H10) there is some σ′_{i}, 0 < σ′_{i} < 1, such that : L^{2}(Ω) ⊂ W^{−σ′_{i},p′_{i}}(Ω) where p′_{i} is the conjugate of p_{i} : that is , \frac{1}{p_{i}} + \frac{1}{p′_{i}} = 1 Simon’s regularity results [18], concerning the problem

−△_{p_{i}}u_{i} = −f_{i}(., u) − b_{i}(u_{i})_{t} ∈ L^{∞}(τ, +∞; B_{∞}^{−σ′_{i},p′_{i}} (Ω)).

Then give for any t ≥ τ,

∥u_{i}(., t)∥_{B_{∞}^{1+(1−σ′_{i})(1−p_{i})^{2},p_{i}} (Ω)} ≤ c_{35} ∥f_{i}(., w) − b′_{i}(u_{i}) (u_{i})_{t}∥_{B_{∞}^{−σ′_{i},p{′ \atop i}} (Ω)} + c_{36}(τ).

Hence estimate (4.17) follows. For a solution (u_{1}, u_{2}) of (S), we define the ω − limit set by : ω(φ_{1}, ψ_{1}) =

w = (w_{1}, w_{2}) ∈ (W_{0}^{1,p_{1}}(Ω) × L^{∞}(Ω)) ∩ (W_{0}^{1,p_{2}}(Ω) × L^{∞}(Ω))

∃t_{n} → +∞ | \begin{matrix}u_{1}(., t_{n}) → w_{1} in L^{p_{1}}(Ω) \\ u_{2}(., t_{n}) → w_{2} in L^{p_{2}}(Ω) \end{matrix}

Corollary 4.1 Under the assumptions (H1) to (H10), we have ω(φ_{1}, ψ_{1}) ≠ ∅ and any (w_{1}, w_{2}) ∈ ω(φ_{1}, ψ_{1}) is a bounded weak solution of the stationary problem

\{ \begin{matrix}−Δ_{p_{i}}w_{i} + f_{i}(x, w) = 0 in Ω \\ w_{i} = 0  on ∂Ω \end{matrix}

Proof. From (4.19) we obtain ω(φ_{1}, ψ_{1}) ≠ ∅, letting w_{i} = lim_{n→+∞} u_{i}(., t_{n}) and

w = (w_{1}, w_{2}) ∈ ω(φ_{1}, ψ_{1}), we get that w is a solution of the Dirichlet problem for elliptic system. The proof is analogous to DIAZ and DE THELIN [4] and is omitted.

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Hamid El Ouardi Ecole Nationale Supérieure d’Electricité et de Mécanique B.P. 8118 -Casablanca-Oasis, Maroc and UFR Mathématiques Appliquées et Industrielles Faculté des Sciences, El Jadida - Maroc E-mail adress: helou_{−}di@yahoo.fr, elouardi@ensem − uh2c.ac.ma Abderrahmane El Hachimi UFR Mathématiques Appliquées et Industrielles Faculté des Sciences B.P. 20, El Jadida - Maroc E-mail adress: elhachimi@ucd.ac.ma

(Received April 19, 2004)