Existence and Attractors of Solutions for Nonlinear Parabolic Systems

By A. El Hachimi and H. El Ouardi

Abstract: We prove existence and asymptotic behaviour results for weak solutions of a mixed problem (S). We also obtain the existence of the global attractor and the regularity for this attractor in [H^{2}(Ω)]^{2} and we derive estimates of its Haussdorf and fractal dimensions.

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Keywords: Nonlinear parabolic systems; existence of solutions; global attractor; asymptotic behaviour; Haussdorf and fractal dimensions.

AMS subject classifications : 35K55, 35K57, 35K65, 35B40

0. Introduction

We consider the following nonlinear system

(S) \{ \begin{matrix}\frac{∂b_{1}(u_{1})}{∂t} − △u_{1} + f_{1}(x, u_{1}, u_{2}) = 0 in Ω × (0, T) \\ \frac{∂b_{2}(u_{2})}{∂t} − △u_{2} + f_{2}(x, u_{1}, u_{2}) = 0 in Ω × (0, T) \end{matrix} u_{1} = u_{2} = 0 in ∂Ω × (0, T) (b_{1}(u_{1}(x, 0), b_{2}(u_{2}(x, 0)) = (b_{1}(φ_{0}(x)), b_{2}(ψ_{0}(x))) in Ω

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 \frac{∂β(u)}{∂t} − Δu + 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.

1. 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) = ∫_{0}^{t} b_{i}(τ)dτ . Then the Legendre transform Ψ^{∗} of Ψ is defined by

Ψ_{i}^{∗}(τ) = sup_{s∈R} {τs − Ψ_{i}(s)} . Ω stands for 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), 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:

(H1) (φ_{0}, ψ_{0}) ∈ L^{2}(Ω) × L^{2}(Ω). (H2) b_{i} is an increasing continuous function from R into R , b_{i}(0) = 0, and

there exists c_{ij} > 0 such that : |b_{i}(s)| ≤ c_{i1} |s| + c_{i2}, for all s ∈ R, i = 1, 2. ( H3) f_{i} ∈ C^{1}(\overline{Ω} × R × R). ( H4) ∀x ∈ Ω, ∀ξ ∈ R, ∃c_{1} > 0, c_{2} > 0 :

\{ \begin{matrix}sign(ξ)f_{1}(x, ξ, 0) ≥ −c_{1} \\ sign(ξ)f_{2}(x, 0, ξ) ≥ −c_{2}. \end{matrix}

( H5) For any N > 0, ∃c_{3} > 0, c_{4} > 0, c_{5} > 0 :

sign(ξ)f_{1}(x, ξ, v) ≥ c_{3} |ξ|^{p_{1}−1} − c_{4}

|f_{1}(x, ξ, v)| ≤ c_{5}(|ξ|^{p_{1}−1} + 1) p_{1} > 2

|f(x, u, v)| ≤ a_{1}(|u|), where a : R^{+} → R^{+} is increasing

for any v : |v| ≤ N.

( H6) For any M > 0, ∃c_{6} > 0, c_{7} > 0, c_{8} > 0 :

sign(ξ)f_{2}(x, u, ξ) ≥ c_{6} |ξ|^{p_{2}−1} − c_{7}

|f_{2}(x, u, ξ)| ≤ c_{8}(|ξ|^{p_{2}−1} + 1) p_{2} > 2

|f_{2}(x, u, v)| ≤ a_{2}(|v|), where a_{2} : R^{+} → R^{+} is increasing

for any u : |u| ≤ M.

(H7) 0 < γ_{i} ≤ b′_{i}(s) for all s ∈ R.

Definition By a weak solution of (S), we mean an element u_{i} ∈ L^{p_{i}}(0, T; L^{p_{i}}(Ω)) ∩ L^{2}(0, T; H_{0}^{1}(Ω)) ∩ L^{∞}(t_{0}, T; L^{∞}(Ω)), for all t_{0} > 0

such that \frac{∂b_{i}(u_{i})}{∂t} ∈ L^{p_{i}^{∗}}(0, T; L^{p_{i}^{∗}}(Ω))+L^{2}(0, T; H^{−1}(Ω)) and ∀ϕ_{i} ∈ L^{2}(0, T; H^{−1}(Ω)) :

∫_{0}^{T} ⟨\frac{∂b_{i}(u_{i})}{∂t}, ϕ_{i}⟩_{V_{i}^{∗},V_{i}} dt + ∫_{0}^{T} ∫_{Ω} ∇u_{i}∇ϕ_{i}dxdt + ∫_{0}^{T} ∫_{Ω} f_{i}(x, u_{1}, u_{2})dxdt = 0,

and if (ϕ_{i})_{t} ∈ L^{2}(0, T; L^{2}(Ω)), ϕ_{i}(T) = 0 ∫_{0}^{T} ⟨\frac{∂b_{i}(u_{i})}{∂t}, ϕ_{i}⟩_{V′_{i} ,V_{i}} dt = − ∫_{0}^{T} ∫_{Ω} (b_{i}(u_{i}(t) − b_{i}(u_{i}(x, 0)) (ϕ_{i})_{t}dxdt,

where V_{i} = L^{p_{i}}(Ω) ∩ H_{0}^{1}(Ω), Vi^{′} = L^{p′_{i}}(Ω) + H^{−1}(Ω), _{\frac{1}{p′_{i}}} + \frac{1}{p_{i}} = 1, i = 1, 2.

1.2. Existence theorem. 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 u_{i} ∈ L^{p_{i}}(0, T; L^{p_{i}}(Ω)) ∩ L^{2}(0, T; H_{0}^{1}(Ω)) ∩ L^{∞}(t_{0}, T; L^{∞}(Ω)), ∀t_{0} > 0 Proof: By theorem 3.2 in [8] ,we can choose u_{i}^{0} ∈ L^{p_{i}}(Q_{T})∩L^{2}(0, T; H_{0}^{1}(Ω))∩

L^{∞}(τ, T; L^{∞}(Ω)), for any τ > 0 such that :

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

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

and we construct two sequences of functions (u_{1}^{n}) and (u_{2}^{n}), such that :

\{ \begin{matrix}\frac{∂b_{1}(u_{1}^{n})}{∂t} − △u_{1}^{n} + f(x, u_{1}^{n}, u_{2}^{n−1}) = 0  in Q_{T}  (1.1) \\ u_{1}^{n} = 0  in S_{T}  (1.2) \\ b_{1}(u_{1}^{n})_{t=0} = b_{1}(φ_{0})  in Ω  (1.3) \end{matrix}

\{ \begin{matrix}\frac{∂b_{2}(u_{2}^{n})}{∂t} − △u_{2}^{n} + f_{2}(x, u_{1}^{n−1}, u_{2}^{n}) = 0  in Q_{T}  (1.4) \\ u_{2}^{n} = 0  in S_{T}  (1.5) \\ b_{2}(u_{2}^{n})_{t=0} = b_{2}(ψ_{0})  in Ω  (1.6) \end{matrix}

We need lemma 1 and lemma 2 below to complete the proof of theorem 1. From now on we denote by c_{i} various positive constants independent of n. Lemma 1

∀τ > 0, ∃c_{τ} > 0 such that ∥u_{i}^{n}∥_{L^{∞}(τ,T;L^{∞}(Ω))} ≤ c_{τ} .  (1.7)

Proof : For n = 0, (1.7) is proved in [7]. So, suppose (1.7) for (n − 1). Multiplying (1.1) by |b_{1}(u_{1}^{n})|^{k} b_{1}(u_{1}^{n}) and using (H2),(H5) , we obtain :

\frac{1}{k + 2} ∫_{Ω} |b_{1}(u_{1}^{n})|^{k+2} dx + c_{9} ∫_{Ω} |b_{1}(u_{1}^{n})|^{k+p_{1}} dx ≤ c_{10} ∫_{Ω} |b_{1}(u_{1}^{n})|^{k+1} dx.

Setting y_{k,n}(t) = ∥b_{1}(u_{1}^{n})∥_{L^{k+2}(Ω)} and using Holder’s inequality on both sides, we have the existence of two constants λ > 0 and δ > 0 such that

\frac{dy_{k,n}(t)}{dt} + λy_{k,n}^{p_{1}−1}(t) ≤ δ,

which implies from lemma 5.1([22]) that ∀t ≥ τ > 0

y_{k,n}(t) ≤ (\frac{δ}{λ})^{\frac{1}{p_{1}−1}} + \frac{1}{[λ(p_{1} − 2)t]^{\frac{1}{p_{1}−2}}} .

As k → ∞, we obtain

|u_{1}^{n}(t)| ≤ c_{τ} ∀t ≥ τ > 0 .

The same holds also for u_{2}^{n}.■

Lemma 2 ∀τ > 0, ∃c_{i} = c_{i} (τ, φ_{0}, ψ_{0}) > 0 :

∥u_{i}^{n}∥_{L^{2}(0,T;H_{0}^{1}(Ω))} ≤ c_{11,}
∥u_{i}^{n}∥_{L^{∞}(τ,T;H_{0}^{1}(Ω)∩L^{∞}(Ω))} ≤ c_{12}

and

2 ; ∑_{i=1} [∫_{0}^{T} ∫_{Ω} |∇u_{i}^{n}|^{2} dx + c_{13} ∫_{0}^{T} ∫_{Ω} |u_{i}^{n}|^{p_{i}} dx] ≤ c_{14}.

Proof of lemma 2 : Multiplying (1.1) by u_{1}^{n} and (1.4) by u_{2}^{n}, and adding , we get :

\frac{d}{dt} 2 ; ∑_{i=1} [∫_{Ω} Ψ_{i}^{∗}(b_{i}(u_{i}^{n}))dx] + 2 ; ∑_{i=1} ∫_{Ω} |∇u_{i}^{n}|^{2} dx + c_{15} 2 ; ∑_{i=1} ∫_{Ω} |u_{i}^{n}|^{p_{i}} dx ≤ c_{16}. (1.8)

But |φ_{0}|_{L^{2}(Ω)} + |ψ_{0}|_{L^{2}(Ω)} ≤ c ⇒ ∫_{Ω} Ψ_{1}^{∗}(b_{1}(φ_{0}))dx + ∫_{Ω} Ψ_{2}^{∗}(b_{2}(ψ_{0}))dx ≤ c,

so we deduce that 2 ; ∑_{i=1} ∫_{0}^{T} ∫_{Ω} |∇u_{i}^{n}|^{2} dx + c_{17} 2 ; ∑_{i=1} ∫_{0}^{T} ∫_{Ω} |u_{i}^{n}|^{p_{i}} dx ≤ c_{18}.

Whence lemma 2. From lemma 2 and Lemma 1, there is a subsequence u_{i}^{n} (i = 1, 2) with the

following properties:

u_{i}^{n} → u_{i} weakly in L^{2}(0, T; H_{0}^{1}(Ω))∩ L^{p_{i}}(0, T; L^{p_{i}}(Ω)) b_{i}(u_{i}^{n}) → χ_{i}weakly in L^{2}(0, T; L^{2}(Ω)) b_{i}(u_{i}^{n}) → χ_{i}strongly in L^{2}(τ, T; H^{−1}(Ω)) ( by the compactness result of

Aubin ( see [22])). By lemma 7([9]), we have χ_{i} = b_{i}(u_{i}). Moreover, f_{1}(., u_{1}^{n}, u_{2}^{n−1}) converges to f_{1}(., u_{1}, u_{2}) in L^{r}(τ, T; L^{r}(Ω)), ∀r ≥ 1, ∀τ ≥ 1 and f_{2}(., u_{1}^{n−1}, u_{2}^{n}) converges to f_{2}(., u_{1}, u_{2}) in L^{r}(τ, T; L^{r}(Ω)), ∀r ≥ 1; taking the limit as n goes to ∞ , we deduce that ( u_{1}, u_{2}) is a weak solution

of (S).

1.3. Uniqueness Theorem 2. Assume that f_{1} and f_{2} verify :

(H8)\{ \begin{matrix}∀M > 0, ∀N > 0, ∃c_{M} > 0, c_{N} > 0 : \\ ∀u, \overline{u}, v, \overline{v} : |u| + |\overline{u}| ≤ M and |v| + |\overline{v}| ≤ N, we have \\ |f_{1}(x, u, v) − f_{1}(x, \overline{u}, \overline{v})|^{2} + |f_{2}(x, u, v) − f_{2}(x, \overline{u}, \overline{v})|^{2} ≤ \\ c_{M}(b_{1}(u) − b_{1}(\overline{u}))(u − \overline{u}) + c_{N}(b_{2}(v) − b_{2}(\overline{v}))(v − \overline{v}). \end{matrix}

Then (S) has a unique solutions (u, v) in Q_{T} . Proof : Let (u, v) and ( \overline{u}, \overline{v}) be solutions of (S); then we have :

\frac{∂(b_{1}(u) − b_{1}(\overline{u}))}{∂t} − △(u − \overline{u}) = f_{1}(x, \overline{u}, \overline{v}) − f_{1}(x, u, v)  (1.9)

and

\frac{∂(b_{2}(v) − b_{2}(\overline{v}))}{∂t} − △(v − \overline{v}) = f_{2}(x, \overline{u}, \overline{v}) − f_{2}(x, u, v).  (1.10)

Multiplying (1.9) by w_{1} = (−△)^{−1}(b_{1}(u) − b_{1}(\overline{u})) and (1.10) by w_{2} = (−△)^{−1}(b_{2}(v) − b_{2}(\overline{v})) and adding, we get \frac{1}{2} \frac{d}{dt} [∥b_{1}(u) − b_{1}(\overline{u})∥_{H^{−1}(Ω)}^{2} + ∥b_{2}(v) − b_{2}(\overline{v})∥_{H^{−1}(Ω)}^{2}] +

(b_{1}(u) − b_{1}(\overline{u}), u − \overline{u})_{L^{2}(Ω)} + (b_{2}(v) − b_{2}(\overline{v}), v − \overline{v})_{L^{2}(Ω)} ≤
c ∥f_{1}(x, u, v) − f_{1}(x, \overline{u}, \overline{v})∥_{H^{−1}(Ω)} ∥b_{1}(u) − b_{1}(\overline{u})∥_{H^{−1}(Ω)} +

c ∥f_{2}(x, u, v) − f_{2}(x, \overline{u}, \overline{v})∥_{H^{−1}(Ω)} ∥b_{2}(v) − b_{2}(\overline{v})∥_{H^{−1}(Ω)} .  (1.11)

From hypothesis (H8) we obtain

∥f_{1}(x, u, v) − f_{1}(x, \overline{u}, \overline{v})∥_{H^{−1}(Ω)}^{2} + ∥f_{2}(x, u, v) − f_{2}(x, \overline{u}, \overline{v})∥_{H^{−1}(Ω)}^{2}

≤ c [∥f_{1}(x, u, v) − f_{1}(x, \overline{u}, \overline{v})∥_{L^{2}(Ω)}^{2} + ∥f_{2}(x, u, v) − f_{2}(x, \overline{u}, \overline{v})∥_{L^{2}(Ω)}^{2}]

≤ cc_{M}(b_{1}(u) − b_{1}(\overline{u}), u − \overline{u})_{L^{2}(Ω)} + cc_{N}(b_{2}(v) − b_{2}(\overline{v}), v − \overline{v})_{L^{2}(Ω)} , (1.12)

where M = ∥u∥_{L^{∞}(0,T;L^{∞}(Ω))}+∥\overline{u}∥_{L^{∞}(0,T;L^{∞}(Ω))} and N = ∥v∥_{L^{∞}(0,T;L^{∞}(Ω))}+ ∥\overline{v}∥_{L^{∞}(0,T;L^{∞}(Ω))} .

Therefore, using Schwartz inequality in (1.11), the fact that ( b_{i}, i = 1, 2 ) is increassing and (1.12), we deduce that

\frac{d}{dt} [∥b_{1}(u) − b_{1}(\overline{u})∥_{H^{−1}(Ω)}^{2} + ∥b_{2}(v) − b_{2}(\overline{v})∥_{H^{−1}(Ω)}^{2}] ≤

c\frac{d}{dt} [∥b_{1}(u) − b_{1}(\overline{u})∥_{H^{−1}(Ω)}^{2} + ∥b_{2}(v) − b_{2}(\overline{v})∥_{H^{−1}(Ω)}^{2}] .

Thus, we deduce that b_{1}(u) = b_{1}(\overline{u}) and b_{2}(v) = b_{2}(\overline{v}), hence u = \overline{u} and v = \overline{v}.

Remark 1. Theorem 1 establishes the existence of dynamical system {S(t)}_{t≥0} which maps L^{2}(Ω)×L^{2}(Ω) into L^{2}(Ω)×L^{2}(Ω) such that S(t)(φ_{0}, ψ_{0}) = (u_{1}(t), u_{2}(t)).

2. Global attractor

Proposition 1 Assume that (H1)-( H8) hold; then the solution (u_{1}, u_{2}) of system (S) satisfies :

|u_{1}(t)|_{L^{∞}(Ω)} + |u_{2}(t)|_{L^{∞}(Ω)} ≤ c(t_{0}) ∀t ≥ t_{0}  (2.0)

and

2 ; ∑_{i=1} ∫_{Ω} |∇u_{i}|^{2} dx ≤ c ∀t ≥ t_{0} + r  (2.1)

Proof : Reasoning as the proof of lemma 1, we also have(2.0). Multiplying the first equation of (S) by u_{1} and the second by u_{2}, by (H2)

and (2.5), we get :

\frac{d}{dt} 2 ; ∑_{i=1} ∫_{Ω} Ψ_{i}^{∗}(b_{i}(u_{i}))dx + 2 ; ∑_{i=1} \frac{1}{2} ∫_{Ω} |∇u_{i}|^{2} dx = − 2 ; ∑_{i=1} ∫_{Ω} f_{i}(x, u)u_{i}dx ≤ c. (2.2)

For fixed r > 0 and τ > 0, integrate (2.2) on ]t, t + r[

∀t ≥ τ > 0 2 ; ∑_{i=1} ∫_{t}^{t+r} ∫_{Ω} |∇u_{i}|^{2} dxds ≤ c(τ).  (2.3)

Multiplying the first equation of (S) by (u_{1})_{t} and the second by (u_{2})_{t}, we get

\frac{1}{2} \frac{d}{dt}( 2 ; ∑_{i=1} ∫_{Ω} |∇u_{i}|^{2} dx) + 2 ; ∑_{i=1} ∫_{Ω}(b′_{i}(u_{i})(\frac{∂u_{i}}{∂t})^{2}dx =  (2.4)

2 ; ∑_{i=1} ∫_{Ω} f_{i}(x, u)(u_{i})_{t} ≤ \frac{1}{2} ∑_{i=1}^{2} ∫_{Ω} \frac{f_{i}^{2}(x, u)}{b′_{i}(u_{i})}dx + \frac{1}{2} 2 ; ∑_{i=1} ∫_{Ω}(b′_{i}(u_{i})(\frac{∂u_{i}}{∂t})^{2}dx.

By (H7) and the properties of functions f_{i} , we obtain:

\frac{d}{dt} [ 2 ; ∑_{i=1} ∫_{Ω} |∇u_{i}|^{2} dx] ≤ c(τ).  (2.5)

From the uniform Gronwall’s lemma see [22], we get (2.1).■

Remark 2. By proposition1 we deduce that there exist absorbing sets in L^{σ^{1}}(Ω) × L^{σ^{1}}(Ω) for any σ_{i} : 1 ≤ σ_{i} ≤ +∞ and absorbing sets in H_{0}^{1}(Ω) × H_{0}^{1}(Ω); then assumptions (1.1) ,(1.4) and (1.12) in theorem 1.1 [22, p.23] are satisfied with U = [L^{2}(Ω)]^{2} , so we have the following :

Theorem 2. Assume that (H1) - (H7) are satisfied. Then the semi-group S(t) associated with the boundary value problem (S) possesses a maximal attractor A, which is bounded in [H_{0}^{1}(Ω) ∩ L^{∞}(Ω)] × [H_{0}^{1}(Ω) ∩ L^{∞}(Ω)] , compact and connected in [L^{2}(Ω)]^{2} . Its domain of attraction is the whole space [L^{2}(Ω)]^{2} .

3. 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

(H9) {N ≤ 3 and b_{i} is of class C^{3}.

Hereafter, we shall assume that there exist positive constants δ_{i} > 0 and a function Φ from R^{N+2} to R such that :

(H10) \{ \begin{matrix}f_{i}(x, u) = f_{i}(u) − h_{i}(x) = δ_{i} \frac{∂Φ}{∂u_{i}} \\ f_{i} satisfying (H3) to (H6) and h_{i} ∈ L^{∞}(Ω). \end{matrix}

We shall denote : r(t) = 2 ; ∑_{i=1} ∫_{Ω} b′_{i}(u_{i}) (ui^{′})^{2} dx.

Theorem 3 Let f_{i} and b_{i} satisfies hypothesis (H1) to (H10). Then the solution ( u_{1}, u_{2}) of problem (S) satisfies the following regularity estimates:

\frac{∂ b_{i}(u_{i})}{∂t} ∈ L^{2}(t_{0}, +∞; L^{2}(Ω) ),  (3.0)

\frac{∂ ∇u_{i}}{∂t} ∈ L^{2}(t_{0}, +∞; L^{2}(Ω) ),  (3.1)

and

u_{i} ∈ H^{2} (Ω) .  (3.2)

To prove this theorem, we need the following lemma: Lemma 3 Under the assumptions of theorem 3, there exist constants C =

C(φ_{0}, ψ_{0}), such that for any T > 0 :

∥u_{i}∥_{L^{∞}(0,T,H_{0}^{1}(Ω))} ≤ C < ∞  (3.3)

and

∥\frac{∂u_{i}}{∂t} ∥_{L^{2}(Q_{T})} ≤ C < ∞.  (3.4)

Proof of lemma 3 : Multiplying the equation \frac{∂b_{i}(u_{i})}{∂t} −div [∇u_{i}]+δ_{i} \frac{∂Φ}{∂u_{i}} = 0 by \frac{1}{δ_{i}} (u_{i})_{t} and adding the two equations, we obtain :

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

∫_{Ω} [−Φ(., u_{1}(T), u_{2}(T)) + Φ(., φ_{0}, ψ_{0}] dx = \frac{1}{2δ_{1}} ∫_{Ω} |∇φ_{0}|^{2} dx + \frac{1}{2δ_{2}} ∫_{Ω} |∇ψ_{0}|^{2} dx. (3.5)

Φ being continuous and (u_{1}, u_{2}) bounded, we then obtain:

2 ; ∑_{i=1} \begin{matrix}γ & i \\ δ & i \end{matrix} ∫_{Q_{T}} (\frac{∂u_{i}}{∂t})^{2}dxdt + 2 ; ∑_{i=1} \frac{1}{2δ_{i}} ∫_{Ω} |∇u_{i}(., T)|^{2} dx ≤ C(φ_{0}, ψ_{0}) ,  (3.6)

whence (3.3) and (3.4).■ Proof of theorem 3 : Differentiating equation \frac{∂b_{i}(u_{i})}{∂t} − div [∇u_{i}] +

f_{i}(u_{1}, u_{2}) = h_{i} we obtain

b′_{i}(u_{i})u′′_{i} + b′′_{i}(u_{i})(u′_{i})^{2} − div (∇u_{i}))′ + 2 ; ∑_{j=1} \frac{∂f_{i}(u)}{∂u_{j}}u′_{j} = 0 .  (3.7)

Now multiplying (3.7) by u′_{i}, and integrating over Ω gives

\frac{1}{2}r′(t) + \frac{1}{2} 2 ; ∑_{i=1} ∫_{Ω} b′′_{i}(u_{i})(u′_{i})^{3}dx + 2 ; ∑_{i=1} ∥u_{i}′∥_{H_{0}^{1}(Ω)} + ∫_{Ω} ( 2 ; ∑_{i=1} 2 ; ∑_{j=1} \frac{∂f_{i}(u)}{∂u_{j}}u′_{j}) u′_{i}dx = 0 . (3.8)

The L^{∞} estimate and hypothesis ( H9) imply successively :

∫_{Ω} ( 2 ; ∑_{i=1} 2 ; ∑_{j=1} \frac{∂f_{i}(u)}{∂u_{j}}u′_{j}) u′_{i}dx ≤ M 2 ; ∑_{i=1} ∫_{Ω} (u′_{i})^{2} dx,  (3.9)

γ 2 ; ∑_{i=1} ∫_{Ω} (u′_{i})^{2} d ≤ r(t) ≤ M 2 ; ∑_{i=1} ∥u_{i}′∥_{H_{0}^{1}(Ω)}^{2} ,  (3.10)

and

−\frac{1}{2} 2 ; ∑_{i=1} ∫_{Ω} b′′(u_{i})(u′_{i})^{3}dx ≤ 2 ; ∑_{i=1} \frac{M_{i}}{2} |u′_{i}|_{L^{3}(Ω)}^{3} .  (3.11)

Since for N ≤ 3, H_{0}^{1}(Ω) is continuously imbedded in L^{6}(Ω), we then obtain by Young’s inequality that :

|u_{i}′|_{L^{3}(Ω)}^{3} ≤ c |u′_{i}|L^{2}(Ω)^{\frac{9}{4}} |u′_{i}|_{L^{3}(Ω)}^{3} ≤ c |u′_{i}|L^{2}(Ω)^{\frac{18}{5}} + \frac{1}{2} ∥u_{i}′∥_{H_{0}^{1}(Ω)}^{2} .  (3.12)

By (3.9),(3.10),(3.11) and (3.12), (3.7) becomes :

r′(t) + \frac{1}{2} 2 ; ∑_{i=1} ∥u_{i}′∥_{H_{0}^{1}(Ω)}^{2} + cr(t) ≤ cr(t)^{\frac{9}{5}} + cr(t) ≤ cr(t)^{2} + c.  (3.13)

On the other hand, using (2.4) we obtain :

2 ; ∑_{i=1} ∫_{τ}^{τ+r} ∫_{Ω} b′_{i}(u_{i}) (u′_{i})^{2} dxdt ≤ c_{τ}, for any τ ≥ t_{0} .  (3.14)

Estimates (3.13) and (3.14) and the use of the uniform Gronwall’ lemma thus gives

r(t) ≤ c(t_{0}), for any  ∀t ≥ t_{0}.  (3.15)

Now, by (3.15) and hypothesis (H1) , we get :

2 ; ∑_{i=1} ∫_{Ω} (\frac{∂b_{i}(u_{i})}{∂t})^{2} dx ≤ Mr(t) ≤ c(t_{0}) for any t ≥ t_{0} .

Then (3.0) is satisfied. Now, as we have : −△u_{i} = −f_{i}(x, u) − b_{i}(u_{i})_{t} ∈ L^{∞}(t_{0}, +∞; L^{2} (Ω)), then by (S) u_{i}(., t) is in bounded subset of H^{2}(Ω). Hence estimate (3.2)

follows.■

For a solution (u_{1}, u_{2}) of (S), we define the ω − limit set by :

ω(φ_{0}, ψ_{0}) = \{ \begin{matrix}w = (w_{1}, w_{2}) ∈ (H_{0}^{1}(Ω) × L^{∞}(Ω)) ∩ (H_{0}^{1}(Ω) × L^{∞}(Ω)) \\ ∃t_{n} → +∞ u_{1}(., t_{n}) → w_{1} in L^{2}(Ω), u_{2}(., t_{n}) → w_{2} in L^{2}(Ω) \end{matrix} \} .

Corollary 1. Under the assumptions (H1) to (H10), we have ω(φ_{0}, ψ_{0}) ≠ ∅ and any (w_{1}, w_{2}) ∈ ω(φ_{0}, ψ_{0}) is a bounded weak solution of the stationary problem

\{ \begin{matrix}−Δu_{i} + f_{i}(x, u_{1}, u_{2}) = 0 in Ω \\ u_{i} = 0 on  ∂Ω \end{matrix}

Proof : From (3.3) we obtain ω(φ_{0}, ψ_{0}) ≠ ∅. Setting w_{i} = Lim_{n→∞} u_{i}(., t_{n}) and w = (w_{1}, w_{2}) ∈ ω(φ_{0}, ψ_{0}), we get that w = (w_{1}, w_{2}) is a solution of the Dirichlet problem for elliptic system. The proof is analogous to El Ouardi and de Thélin [12] and is omitted here.

Corollary 2. Under the assumptions (H1) to (H10), we have

A ⊂ (W^{2,6}(Ω))^{2} if N = 3

and

A ⊂ (W^{2,r}(Ω))^{2} for all r < ∞ if N ≤ 2.

Proof: Taking the inner product of (4.7) with u′′_{i}, we get

\frac{d}{dt}( 2 ; ∑_{i=1} ∥u′_{i}∥2_{H^{1}_{0}(Ω)}) ≤ c( 2 ; ∑_{i=1} |u′_{i}|4_{H^{1}_{0}(Ω)} + 2 ; ∑_{i=1} |u′_{i}|2_{L^{2}(Ω)}).

By uniform Gronwall’s lemma, we get 2 ; ∑_{i=1} ∥u′_{i}∥2_{H^{1}_{0}(Ω)} ≤ c, ∀t ≥ T.

Then 2 ; ∑_{i=1} ∥u′_{i}∥2_{L^{α}i (Ω)} ≤ c, ∀t ≥ T for all t ≥ τ and α_{i} = 6 if N = 3 or

1 ≤ α_{i} < ∞ if N ≤ 2.

4. Dimension of the attractor A

4.1 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

(S_{L})  \{ \begin{matrix}U = (U_{1}, U_{2}) ∈ [L^{2}(0, T; H_{0}^{1}(Ω)]^{2} \\ \frac{∂}{∂t} (b′_{i}(u_{i}U_{i}) − △U_{i} + 2 ; ∑_{j=1} \frac{∂f_{i}}{∂u_{j}} U_{j} = 0 \\ U(0) = (U_{1}(0), U_{2}(0)) = U_{0} \end{matrix}

The existence and uniqueness of solution can be deduced from (4.0) below

U_{i} ∈ L^{∞}(0, T; L^{2}(Ω)) ∩ L^{2}(0, T; H_{0}^{1}(Ω)).  (4.0)

To deduce (4.0), Multiply the equation in (S_{L}) by b′_{i}(u_{i})U_{i}, we obtain

\frac{1}{2} \frac{d}{dt}( 2 ; ∑_{i=1} |b′_{i}(u_{i})U_{i}|_{L^{2}(Ω)}^{2}) + 2 ; ∑_{i=1} (∇u_{i}, ∇(b′_{i}(u_{i})U_{i}))_{L^{2}(Ω)}

= 2 ; ∑_{i=1}( 2 ; ∑_{j=1} \frac{∂f_{i}}{∂u_{j}} U_{j}, b′_{i}(u_{i})U_{i})_{L^{2}(Ω)}  (4.1)

By the hypothesis on b_{i}, we have

∇(b′_{i}(u_{i})U_{i}) = b′_{i}(u_{i})∇U_{i} + b′′_{i}(u_{i})∇u_{i}.U_{i},  (4.2)

(∇U_{i}, b′′_{i}(u_{i})∇u_{i}.U_{i})_{L^{2}(Ω)} ≤ c |∇u_{i}|_{L^{4}(Ω)} |U_{i}|_{L^{4}(Ω)} |∇U_{i}|_{L^{2}(Ω)} ,  (4.3)

and

2 ; ∑_{i=1}( 2 ; ∑_{j=1} \frac{∂f_{i}}{∂u_{j}} U_{j}, b′_{i}(u_{i})U_{i})_{L^{2}(Ω)} ≤ M 2 ; ∑_{i=1} 2 ; ∑_{j=1} |U_{j}U_{i}|_{L^{1}(Ω)} ≤ c 2 ; ∑_{i=1} |U_{i}|2_{L^{2}(Ω)} . (4.4)

From (4.2) to (4.4), (4.1) becomes

\frac{1}{2} \frac{d}{dt}( 2 ; ∑_{i=1} |b′_{i}(u_{i})U_{i}|_{L^{2}(Ω)}^{2}) + γ 2 ; ∑_{i=1} |U_{i}|2_{H^{1}_{0}(Ω)} ≤ c ∑_{i=1}^{2} |U_{i}|2_{L^{2}(Ω)} ≤ c 2 ; ∑_{i=1} |b′_{i}(u_{i})U_{i}|_{L^{2}(Ω)}^{2} .

By standard application of Gronwall’s inequality, we get (4.0).

4.2 Differentiability of the Semigroup We assume that f_{i} ∈ C^{2}(R × R) (∀i = 1, 2). Let u_{0} =(φ_{0}, ψ_{0}), v_{0} =(\overline{φ}_{0}, \overline{ψ}_{0}),

S(t) be the solution of (S) and S′(t, u_{0}) the solution of (S_{L}). The results of [6] imply the following proposition :

Proposition 3. Assume (H1) to (H10), then for any (u_{0}, v_{0}) ∈ [L^{∞}(Ω) × H^{2}(Ω)]^{2}, we have |S(t)v_{0} − S(t)u_{0} − S′(t, u_{0})(v_{0} − u_{0})|_{(L^{2}(Ω))^{2}} ≤ c(t)o(|v_{0} − u_{0}|_{(L^{2}(Ω))^{2}} )

We need the lemma 3 for the proof of proposition 3: Let (u_{1}, u_{2}) and (v_{1}, v_{2}) be two solutions of (S) in A with (u_{1}(0), u_{2}(0)) =

(φ_{0}, ψ_{0}) and (v_{1}(0), v_{2}(0)) =(φ_{1}, ψ_{1}). Setting w_{1} = u_{1}−v_{1} and w_{2} = u_{2}−v_{2}, we have

Lemme 3 Assume (H1) to (H10). For all T > 0, there exists c(T) > 0 such that for all t ∈ [0, T] ,

2 ; ∑_{i=1} |w_{i}(t)|_{H_{0}^{1}(Ω)}^{2} ≤ c(T) 2 ; ∑_{i=1} |w_{i}(0)|_{H_{0}^{1}(Ω)}^{2} ,  (4.7)

t 2 ; ∑_{i=1} |w_{i}(t)|_{H_{0}^{1}(Ω)}^{2} ≤ c(T) 2 ; ∑_{i=1} |w_{i}(0)|_{L^{2}(Ω)}^{2} ,  (4.8)

and

2 ; ∑_{i=1} |w′_{i}(t)|_{L^{2}(0,T;L^{2}(Ω))}^{2} ≤ c(T) 2 ; ∑_{i=1} |w_{i}(0)|_{H_{0}^{1}(Ω)}^{2}  (4.9)

Proof : We have

\{ \begin{matrix}\frac{∂b_{i}(u_{i})}{∂t} − △u_{i} + f_{i}(x, u) = 0 \\ u(0) = (u_{1}(0), u_{2}(0)) = (φ_{0},ψ_{0}) \end{matrix}  \{ \begin{matrix}\frac{∂b_{i}(v_{i})}{∂t} − △v_{i} + f_{i}(x, v) = 0 \\ v(0) = (v_{1}(0), v_{2}(0)) = (φ_{1},ψ_{1}) . \end{matrix}

Thus, the difference w_{i} = u_{i} − v_{i} satisfies

b′_{i}(u_{i})w′_{i} − △w_{i} = [b′_{i}(v_{i}) − b′_{i}(u_{i})] v′_{i} + f_{i}(v) − f_{i}(u).  (4.10)

Setting

F_{11} = ∫_{0}^{1} \frac{∂f_{1}}{∂u_{1}} (x, u_{1}+θ(u_{2}−u_{1}), u_{2})dθ, F_{21} = ∫_{0}^{1} \frac{∂f_{1}}{∂u_{2}} (x, u_{1}, u_{2}+θ(u_{2}−u_{1}))dθ, F_{12} = ∫_{0}^{1} \frac{∂f_{2}}{∂u_{1}} (x, u_{1} + θ(u_{2} − u_{1}), u_{2})dθ and F_{22} = ∫_{0}^{1} \frac{∂f_{2}}{∂u_{2}} (x, u_{1}, u_{2} + θ(u_{2} −

u_{1}))dθ, (4.10) becomes

b′_{i}(u_{i})w′_{i} − △w_{i} = [b′_{i}(v_{i}) − b′_{i}(u_{i})] v′_{i} + 2 ; ∑_{i=1}F_{ij}w_{j}.  (4.11)

We multiply (4.11) by \frac{w_{i}}{b′_{i}(u_{i})}

\frac{1}{2} \frac{d}{dt} 2 ; ∑_{i=1} |w_{i}|_{L^{2}(Ω)}^{2} + 2 ; ∑_{i=1}(∇w_{i}, ∇(\frac{w_{i}}{b′_{i}(u_{i})}))_{L^{2}(Ω)} =

2 ; ∑_{i=1}(\frac{b′_{i}(v_{i}) − b′_{i}(u_{i})}{b′_{i}(u_{i})}v′_{i}, w_{i})_{L^{2}(Ω)} + 2 ; ∑_{i=1} 2 ; ∑_{j=1}(F_{ij}w_{j}, \frac{w_{i}}{b′_{i}(u_{i})})_{L^{2}(Ω)}  (4.12)

And parallel to lemma 16 in [8], it is easy to see that (4.7), (4.8) and (4.9) hold.

Proof of proposition 3 : It is similar to the proof for the lemma 15 in [8, p.125] and is omited.

4.3 Dimension Estimates Consider the linearized problem

(S_{L}) \{ \begin{matrix}U′_{i} = F′_{i}(u_{i}(τ))U_{i} in Ω × R^{+} \\ U_{i} = 0 on ∂Ω × R \\ U_{i}(0) = ξ_{i} \end{matrix}

where F′_{i}(u_{i}(τ))U_{i} = \frac{1}{b′_{i}(u_{i}(τ))}△U_{i} − \frac{b′′_{i} (u_{i}(τ))}{b′_{i}(u_{i}(τ))}u′_{i}U_{i} − \frac{1}{b′_{i}(u_{i}(τ))} 2 ; ∑_{j=1} \frac{∂f_{i}}{∂u_{j}} U_{j}.

This problem can be rewritten as

(L) \{ \begin{matrix}U′ = F′(u(τ))U \\ U = 0 \\ U(0) = ξ \end{matrix}

where U = ( \begin{matrix}U_{1} \\ U_{2} \end{matrix} ) , F′(u(τ)) = ( \begin{matrix}F′_{1}(u_{1}(τ)) & 0 \\ 0 & F′_{1}(u_{1}(τ)) \end{matrix} ) .

Let \overline{U}_{1}, ......., \overline{U}_{m} be m solutions of (L) corresponding to the initial data ξ_{,}, ........, ξ_{m} and Q_{m}(τ) be the orthogonal projector in H = L^{2}(Ω)×L^{2}(Ω) such that Q_{m}H ⊂ V = H_{0}^{1}(Ω) × H_{0}^{1}(Ω). If {W^{k} = (w_{1}^{k}, w_{2}^{k}}_{k=1}^{m} is an orthonormal basis of Q_{m}(τ)H; then

T r(F′(u(τ))∘Q_{m}(τ) = m ; ∑_{k=1}(F′(u(τ))W^{k}, W^{k})_{H} = 2 ; ∑_{i=1} m ; ∑_{k=1}(F′_{i}(u_{i}(τ))W_{i}^{k}, W_{i}^{k})_{L^{2}(Ω)}

and (F′_{i}(u_{i}(τ))W_{i}^{k}, W_{i}^{k})_{L^{2}(Ω)} = (△w_{i}^{k}, \frac{w_{i}^{k}}{b′_{i}(u_{i}(τ))})_{L^{2}(Ω)} −(\frac{b′′_{i} (u_{i}(τ))}{b′_{i}(u_{i}(τ))}u′_{i}w_{i}^{k}, w_{i}^{k})_{L^{2}(Ω)} −

(\frac{1}{b′_{i}(u_{i}(τ))} 2 ; ∑_{j=1} \frac{∂f_{i}}{∂u_{j}} w_{j}^{k}, w_{i}^{k})_{L^{2}(Ω)}.

Since u = (u_{1}, u_{2}) ∈ (L^{∞}(0, +∞, L^{∞}(Ω)))^{2}, we have b_{i}(u_{i}), b′_{i}(u_{i}),

bi^{′′}(u_{i}) ∈ (L^{∞}(0, +∞, L^{∞}(Ω)))^{2}, so

2 ; ∑_{i=1}(F′_{i}(u_{i}(τ))W_{i}^{k}, W_{i}^{k})_{L^{2}(Ω)} ≤ 2 ; ∑_{i=1}(△w_{i}^{k}, \frac{w_{i}^{k}}{b′_{i}(u_{i}(τ))})_{L^{2}(Ω)} + c 2 ; ∑_{i=1} ∫_{Ω} |u′_{i}| |w_{i}^{k}|^{2} dx +

∫_{Ω} 2 ; ∑_{i=1} 2 ; ∑_{j=1} \frac{∂f_{i}(x, u)}{∂u_{j}}w_{j}^{k}w_{i}^{k})dx.  (4.13)

Now, we have

2 ; ∑_{i=1}(△w_{i}^{k}, \frac{w_{i}^{k}}{b′_{i}(u_{i}(τ))})_{L^{2}(Ω)} ≤ −c 2 ; ∑_{i=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2} + cJ

where

J = 2 ; ∑_{i=1} ∫_{Ω} |∇w_{i}^{k}| |w_{i}^{k}| |∇u_{i}| dx ≤ c( 2 ; ∑_{i=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2})^{\frac{1}{2}} (∫_{Ω} 2 ; ∑_{i=1} |w_{i}^{k}|^{2} dx)^{\frac{1}{2}}

≤ \frac{c}{2} 2 ; ∑_{i=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2} + c(∫_{Ω} 2 ; ∑_{i=1} |w_{i}^{k}|^{2} dx)  (4.14)

and

∫_{Ω} 2 ; ∑_{i=1} 2 ; ∑_{j=1} \frac{∂f_{i}(u)}{∂u_{j}}w_{j}^{k}w_{i}^{k})dx ≤ c(∫_{Ω} 2 ; ∑_{i=1} |w_{i}^{k}|^{2} dx).  (4.15)

According to (4.14) and (4.15), relation (4.13) becomes

2 ; ∑_{i=1}(F′_{i}(u_{i}(τ))W_{i}^{k}, W_{i}^{k})_{L^{2}(Ω)} ≤ −c 2 ; ∑_{i=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2} + c 2 ; ∑_{i=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2} +

c(∫_{Ω} 2 ; ∑_{i=1} |w_{i}^{k}|^{2} dx) + c 2 ; ∑_{i=1} ∫_{Ω} |u′_{i}| |w_{i}^{k}|^{2} dx  (4.16)

Which leads to

T r(F′(u(τ)) ∘ Q_{m}(τ) ≤ −c_{105} 2 ; ∑_{i=1} m ; ∑_{k=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2} + c 2 ; ∑_{i=1} m ; ∑_{k=1} ∥w_{i}^{k}∥_{H_{0}^{1}(Ω)}^{2} +

c(∫_{Ω} 2 ; ∑_{i=1} m ; ∑_{k=1} |w_{i}^{k}|^{2} dx) + c 2 ; ∑_{i=1} m ; ∑_{k=1} ∫_{Ω} |u′_{i}| |w_{i}^{k}|^{2} dx.  (4.17)

We set ρ(x) = m ; ∑_{k=1} |w^{k}(x)|^{2} = 2 ; ∑_{i=1} m ; ∑_{k=1} |w_{i}^{k}(x)|^{2} and γ(t) = max(|u′_{1}(t)| , |u′_{2}(t)| ,

θ(t) = ∫_{Ω} γ(t)^{\frac{5}{2}} dx. So, we get

2 ; ∑_{i=1} m ; ∑_{k=1} ∫_{Ω} |u′_{i}| |w_{i}^{k}|^{2} dx ≤ ∫_{Ω} γ(t)ρ(x)dx ≤ c [∫_{Ω} γ(t)^{\frac{5}{2}} dx]^{\frac{2}{5}} [∫_{Ω} ρ^{\frac{5}{3}} dx]^{\frac{3}{5}} . (4.18)

Therefore by theorem 4.1 in [22] , there exists c′_{1} > 0 such that :

∫_{Ω} ρ^{\frac{5}{3}} dx ≤ c′_{1} m ; ∑_{k=1} ∥w^{k}(x)∥_{H_{0}^{1}(Ω)}^{2} .  (4.19)

So, we get

2 ; ∑_{i=1} m ; ∑_{k=1} ∫_{Ω} |u′_{i}| |w_{i}^{k}|^{2} dx ≤ c ∫_{Ω} γ(t)^{\frac{5}{2}} dx + \frac{c}{5} m ; ∑_{k=1} ∥w^{k}(x)∥_{H_{0}^{1}(Ω)}^{2} .

From (4.18) to (4.19), (4.16) becomes

T r(F′(u(τ)) ∘ Q_{m}(τ) ≤ −c m ; ∑_{k=1} ∥w^{k}∥_{H_{0}^{1}(Ω)}^{2} + c ∫_{Ω} ρdx + c ∫_{Ω} γ(t)^{\frac{5}{2}} dx,

and as in [22] , we obtain :

T r(F′(u(τ)) ∘ Q_{m}(τ) ≤ −cm^{1+ \frac{2}{N}} + c′m + θ(t).  (4.20)

Setting q_{m}(t) = sup_{u_{0}∈A} sup_{ξ_{i}∈H, |ξ_{i}|≤1}_{i=1,....,m} \{\frac{1}{t} ∫_{0}^{t} T r(F′(S(τ)u_{0})) ∘ Q_{m}(τ)dτ\}

and

q_{m} = Lim_{t→+∞} sup q_{m}(t).

Then, by lemma 15 in [8, p.119] , we have ∫_{0}^{η} θdτ ≤ c(η) and u′_{i} ∈ L^{∞}(η, +∞, L^{∞}(Ω)). Thus, q_{m}(t) ≤ \frac{c(η)}{t} − cm^{1+ \frac{2}{N}} + c′m + c′(η) and q_{m} ≤ −cm^{1+ \frac{2}{N}} + c′m + c′(η)

and for all integers j > 0, we get μ_{1} + μ_{2}......... + μ_{j} ≤ q_{j} ≤ −cj^{1+ \frac{2}{N}} + c′j + c′(η).

Hence

μ_{1} + μ_{2}......... + μ_{m} < 0 for any m < c′′.  (4.21)

This shows that the fractal dimension of the attractor A is finite and arguing as for theorem V3.3 in [22], we conclude to the following :

Theorem 4. Assume (H1) to (H10) and let m be an integer satisfying (4.21). Then ( i ) dim_{F ractale}(A) ≤ 2m

( ii ) dim_{H}(A) ≤ m.

Acknowledgement. The authors would like to thank the referee for his refereeing and helpful comments.

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A.El Hachimi: UFR Mathématiques Appliquées et Industrielles Faculté des Sciences, B.P 20, El Jadida - Maroc e-mail adress: aelhachi@yahoo.fr H. El Ouardi: UFR Mathématiques Appliquées et Industrielles Faculté des Sciences, B.P 20, El Jadida - Maroc

and

Ecole Nationale Supérieure d’Electricité et de Mécanique B.P 8118-Casablanca-Oasis - Maroc e mail adress: h.elouardi@ensem-uh2c.ac.ma