Directory UMM :Journals:Journal_of_mathematics:EJQTDE:

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 at-tractor and the regularity for this atat-tractor in

H2(Ω)2

and we derive estimates of its Haussdorf and fractal dimensions.

—————————————–

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

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

0. Introduction

We consider the following nonlinear system

(S)    

  

∂b1(u1)

∂t − △u1+f1(x, u1, u2) = 0 in Ω×(0, T) ∂b2(u2)

∂t − △u2+f2(x, u1, u2) = 0 in Ω×(0, T) u1=u2= 0 in ∂Ω×(0, T) (b1(u1(x,0), b2(u2(x,0)) = (b1(ϕ0(x)), b2(ψ0(x))) in Ω

where Ω is a bounded open subset in RN _{,N ≥1, with a smooth boundary} ∂Ω.(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 wherebi(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.

1. Preliminaries, Existence and Uniqueness

1.1 Notations and Assumptions

Letbi, (i= 1,2) be continuous functions withbi(0) = 0.We define fort∈R

Ψi(t) =

Rt

0bi(τ)dτ .Then the Legendre transform Ψ∗ of Ψ is defined by

Ψ∗

i(τ) = sup s∈R

RN and for any T > 0, we setQT = Ω×(0, T) andST =∂Ω×(0, T), where

∂Ω is the boundary of Ω . The norm in a spaceX will be denoted by : k.k_r

ifX =Lr_{(Ω) for all r : 1≤r≤+∞,k.k}

X otherwise andh., .iX,X′ will denote

the duality product betweenX and its dualX′.

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) (ϕ₀, ψ₀)∈L2_(Ω)×L2_(Ω).

(H2) bi is an increasing continuous function from R into R ,bi(0) = 0,and

there existscij >0 such that : |bi(s)| ≤ci1|s|+ci2,for alls∈R,i= 1,2.

( H3)fi ∈C1(Ω×R×R).

( H4)∀x∈Ω,∀ξ∈R,∃c1>0, c2>0 :

sign(ξ)f1(x, ξ,0)≥ −c1 sign(ξ)f2(x,0, ξ)≥ −c2.

( H5) For anyN >0,∃c3>0, c4>0, c5>0 : 

  

  

sign(ξ)f1(x, ξ, v)≥c3|ξ|p1−1

−c4

|f1(x, ξ, v)| ≤c5(|ξ|p1−1

+ 1) p1>2

|f(x, u, v)| ≤a1(|u|), wherea:R+→R+ is increasing

for anyv:|v| ≤N.

( H6) For anyM >0,∃c6>0, c7>0, c8>0 : 

  

  

sign(ξ)f2(x, u, ξ)≥c6|ξ|p2−1

−c7

|f2(x, u, ξ)| ≤c8(|ξ|p2−1

+ 1) p2>2

|f2(x, u, v)| ≤a2(|v|),wherea2:R+→R+ is increasing

for anyu:|u| ≤M.

(H7) 0< γ_i≤b′_i(s) for all s∈R.

DefinitionBy a weak solution of (S), we mean an element

ui∈Lpi_{(0, T;L}pi_(Ω))∩L2_{(0, T;H}1

0(Ω))∩L∞(t0, T;L∞(Ω)), for allt0 >0

such that

∂bi(ui)

∂t ∈Lp

∗

i(0, T;Lp ∗

i(Ω)) +L2(0, T;H−1(Ω)) and∀φ

i∈L2(0, T;H−1(Ω)) :

RT

0

D

∂bi(ui)

∂t , φi

E

V∗ i,Vi

dt+RT

0

R

Ω∇ui∇φidxdt+

RT

0

R

Ωfi(x, u1, u2)dxdt= 0,

and if (φ_i)t∈L2(0, T;L2(Ω)), φi(T) = 0

RT

0

D_∂b

i(ui)

∂t , φi

E

V′ i,Vi

dt=−RT

0

R

Ω(bi(ui(t)−bi(ui(x,0)) (φi)tdxdt,

whereVi=Lpi(Ω)∩_H1

0(Ω), V

′

i =Lp

′

i(Ω) +H−1(Ω), 1

p′ i

+ 1

pi = 1, i= 1,2.

1.2. Existence theorem.

Theorem 1 Let (H1) to ( H6) be satisfied. Then there exists a solution (u1, u2) of problem (S) such that fori= 1,2 , we have

ui∈Lpi(0, T;Lpi(Ω))∩L2(0, T;H01(Ω))∩L∞(t0, T;L∞(Ω)),∀t0>0

Proof: By theorem 3.2 in [8],we can chooseu0

i ∈Lpi(QT)∩L2(0, T;H01(Ω))∩ L∞(τ , T;L∞(Ω)),for anyτ >0 such that :

 



∂b1(u01)

∂t − △u 0

1+f1(x, u01,0) = 0 in QT u0

1= 0 in ST

b1(u0

 



∂b2(u02)

∂t − △u 0

2+f2(x,0, u02) = 0 in QT

u02= 0 in ST

b1(u0

2)t=0 =b1(ψ0) in Ω

and we construct two sequences of functions (un1) and (un2), such that :

 



∂b1(un1)

∂t − △u n

1+f(x, un1, u2n−1) = 0 inQT (1.1) un

1 = 0 inST (1.2)

b1(un

1)t=0=b1(ϕ0) in Ω (1.3)

 



∂b2(un2)

∂t − △u n

2 +f2(x, un1−1, un2) = 0 inQT (1.4) un

2 = 0 inST (1.5)

b2(un

2)t=0=b2(ψ0) in Ω (1.6)

We need lemma 1 and lemma 2 below to complete the proof of theorem 1. From now on we denote byci various positive constants independent ofn. Lemma 1

∀τ >0,∃cτ >0 such that kun_ik_L∞_{(τ ,T;L}∞_(Ω))≤cτ. (1.7)

Proof : Forn= 0,(1.7) is proved in [7]. So, suppose (1.7) for (n−1).

Multiplying (1.1) by|b1(un 1)|

k b1(un

1) and using (H2),(H5) , we obtain :

1

k+ 2 Z

Ω

|b1(un₁)|k+2dx+c9

Z

Ω

|b1(un₁)|k+p1

dx≤c10

Z

Ω

|b1(un₁)|k+1dx.

Setting yk,n(t) = kb1(un1)kLk+2_(Ω) and using Holder’s inequality on both

sides, we have the existence of two constantsλ >0 andδ >0 such that

dyk,n(t)

dt +λy p1−1

k,n (t)≤δ,

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

yk,n(t)≤(δ

λ)

1

p1−1 + 1

[λ(p1−2)t]p11−2

.

Ask→ ∞,we obtain

|un₁(t)| ≤cτ ∀t≥τ >0.

The same holds also forun2.

Lemma 2 ∀τ >0,∃ci=ci(τ , ϕ0, ψ0)>0 :

kun_ik_L2_(0,T;H1

0(Ω)) ≤ c11,

kun_ik_L∞_{(τ ,T;H}1

0(Ω)∩L∞(Ω)) ≤ c12

and

2

X

i=1

" Z T

0

Z

Ω

|∇un i|

2

dx+c13

Z T

0

Z

Ω

|un i|

pi_dx

#

Proof of lemma 2 : Multiplying (1.1) by un1 and (1.4) byun2,and adding ,

we get :

d dt

2

X

i=1

Z

Ω

Ψ∗_i(bi(un_i))dx

+

2

X

i=1

Z

Ω

|∇un_i|2dx+c15 2

X

i=1

Z

Ω

|un_i|pi

dx≤c16. (1.8)

But

|ϕ0|L2_(Ω)+|ψ₀|_L2_(Ω)≤c⇒

R

ΩΨ

∗

1(b1(ϕ0))dx+

R

ΩΨ

∗

2(b2(ψ0))dx≤c,

so we deduce that

2

P

i=1

RT

0

R

Ω|∇u n i|

2

dx+c17 2

P

i=1

RT

0

R

Ω|u n i|

pi

dx≤c18.

Whence lemma 2.

From lemma 2 and Lemma 1, there is a subsequenceun

i (i= 1,2) with the

following properties:

un

i →ui weakly inL2(0, T;H01(Ω))∩Lpi(0, T;Lpi(Ω)) bi(un

i)→χiweakly inL2(0, T;L2(Ω))

bi(un_i) → χ_istrongly in L2(τ , T;H−1(Ω)) ( by the compactness result of Aubin ( see [22])). By lemma 7([9]), we haveχi=bi(ui).Moreover,

f1(., un1, un2−1) converges tof1(., u1, u2) inLr(τ , T;Lr(Ω)),∀r≥1,∀τ ≥1

andf2(., un₁−1, un

2) converges tof2(., u1, u2) inLr(τ , T;Lr(Ω)),∀r≥1;

taking the limit asngoes to∞, we deduce that (u1, u2) is a weak solution of (S).

1.3. Uniqueness Theorem 2.

Assume thatf1 andf2 verify :

(H8)    

  

∀M >0,∀N >0,∃cM >0, cN >0 :

∀u, u, v, v:|u|+|u| ≤M and |v|+|v| ≤N, we have

|f1(x, u, v)−f1(x, u, v)|2+|f2(x, u, v)−f2(x, u, v)|2≤

cM(b1(u)−b1(u))(u−u) +cN(b2(v)−b2(v))(v−v).

Then (S) has a unique solutions (u, v) inQT.

Proof : Let (u, v) and (u, v) be solutions of (S); then we have :

∂(b1(u)−b1(u))

∂t − △(u−u) =f1(x, u, v)−f1(x, u, v) (1.9)

and

∂(b2(v)−b2(v))

∂t − △(v−v) =f2(x, u, v)−f2(x, u, v). (1.10)

Multiplying (1.9) byw1= (−△)−1(b1(u)−b1(u)) and (1.10) by

w2= (−△)−1_{(b2(v)−b2(v)) and adding, we get} 1

2dtd

h

kb1(u)−b1(u)k2_H−1_(Ω)+kb2(v)−b2(v)k

2 H−1_(Ω)

i +

b1(u)−b1(u), u−u)L2_(Ω)+ (b2(v)−b2(v), v−v

L2_(Ω)≤

ckf2(x, u, v)−f2(x, u, v)k_H−1_(Ω)kb2(v)−b2(v)k_H−1_(Ω). (1.11)

From hypothesis (H8) we obtain

kf1(x, u, v)−f1(x, u, v)k2_H−1_(Ω)+kf2(x, u, v)−f2(x, u, v)k

2 H−1_(Ω)

≤chkf1(x, u, v)−f1(x, u, v)k_L22_(Ω)+kf2(x, u, v)−f2(x, u, v)k

2 L2_(Ω)

i

≤ccM(b1(u)−b1(u), u−u)L2_(Ω)+ccN(b2(v)−b2(v), v−v)_L2_(Ω) , (1.12)

whereM =kuk_L∞_(0,T;L∞_(Ω))+kuk

L∞_(0,T;L∞_(Ω))andN=kvk

L∞_(0,T;L∞_(Ω))+

kvk_L∞_(0,T;L∞_(Ω)).

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

d dt

h

kb1(u)−b1(u)k2_H−1_(Ω)+kb2(v)−b2(v)k

2 H−1_(Ω)

i

≤

cd dt

h

kb1(u)−b1(u)k2_H−1_(Ω)+kb2(v)−b2(v)k

2 H−1_(Ω)

i

.

Thus, we deduce that b1(u) = b1(u) and b2(v) = b2(v), hence u= u and

v=v.

Remark 1. Theorem 1 establishes the existence of dynamical system

{S(t)}_t≥0which mapsL2(Ω)×L2(Ω) intoL2(Ω)×L2(Ω) such thatS(t)(ϕ0, ψ0) =

(u1(t), u2(t)).

2. Global attractor

Proposition 1 Assume that (H1)-( H8) hold; then the solution (u1, u2) of system (S) satisfies :

|u1(t)|_L∞_(Ω)+|u2(t)|

L∞_(Ω)≤c(t0) ∀t≥t0 (2.0)

and

2

X

i=1

Z

Ω

|∇ui|2dx≤c ∀t≥t0+r (2.1)

Proof : Reasoning as the proof of lemma 1, we also have(2.0).

d dt

2

X

i=1

Z

Ω

Ψ∗_i(bi(ui))dx+

2

X

i=1

1 2

Z

Ω

|∇ui|2dx=−

2

X

i=1

Z

Ω

fi(x, u)uidx≤c. (2.2)

For fixedr >0 andτ >0, integrate (2.2) on ]t, t+r[

∀t≥τ >0

2

X

i=1

Z t+r

t

Z

Ω

|∇ui|2dxds≤c(τ). (2.3)

Multiplying the first equation of (S) by (u1)tand the second by (u2)t,we get

1 2

d dt(

2

X

i=1

Z

Ω

|∇ui|2dx) + 2

X

i=1

Z

Ω

(b′_i(ui)( ∂ui

∂t )

2_{dx= (2.4)}

2

X

i=1

Z

Ω

fi(x, u)(ui)t≤ 2

1 2

X

i=1

Z

Ω f2

i(x, u) b′_i(ui) dx+

1 2

2

X

i=1

Z

Ω

(b′i(ui)( ∂ui

∂t ) 2

dx.

By (H7) and the properties of functionsfi , we obtain:

d dt

" ₂ X

i=1

Z

Ω

|∇ui|2dx

#

≤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 Lσ1_(Ω)×Lσ1_{(Ω) for any σ}

i : 1≤σi ≤+∞and absorbing sets in H01(Ω)× H1

0(Ω); then assumptions (1.1) ,(1.4) and (1.12) in theorem 1.1 [22, p.23] are

satisfied with U =

L2_(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

H01(Ω)∩L∞(Ω)

×

H01(Ω)∩L∞(Ω)

, com-pact and connected in

L2_(Ω)2

. Its domain of attraction is the whole space

L2(Ω)2

.

3. A regularity property of the attractor

In this section we shall show supplementary regularity estimates on the so-lution 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)

Hereafter, we shall assume that there exist positive constantsδi >0 and a function Φ fromRN+2 toR such that :

(H10)

fi(x, u) =fi(u)−hi(x) =δi_∂u∂Φ_i fi satisfying (H3) to (H6) andhi∈L∞(Ω).

We shall denote : r(t) =

2

P

i=1

R

Ωb

′

i(ui)

u′_i 2

dx.

Theorem 3 Let fi and bi satisfies hypothesis (H1) to (H10). Then the

solution (u1, u2) of problem (S) satisfies the following regularity estimates:

∂ bi(ui)

∂t ∈L 2₍

t0,+∞;L2(Ω) ), (3.0)

∂∇ui ∂t ∈L

2_(t0,+∞;L2_{(Ω) ), (3.1)}

and

ui∈H2(Ω). (3.2)

To prove this theorem, we need the following lemma:

Lemma 3 Under the assumptions of theorem 3, there exist constantsC=

C(ϕ₀, ψ₀), such that for anyT >0 :

kuikL∞_(0,T,H1

0(Ω)) ≤C <∞ (3.3)

and

∂ui ∂t

L2_(Q T)

≤C <∞. (3.4)

Proof of lemma 3: Multiplying the equation∂bi(ui)

∂t −div[∇ui]+δi ∂Φ ∂ui = 0

by _δ1_i(ui)tand adding the two equations, we obtain :

2

X

i=1

1

δi

Z

QT

b′_i(ui)( ∂ui

∂t )

2_dxdt+ 2

X

i=1

1 2δi

Z

Ω

|∇ui(., T)|2dx=

Z

Ω

[−Φ(., u1(T), u2(T)) + Φ(., ϕ₀, ψ₀]dx= 1 2δ1

Z

Ω

|∇ϕ₀|2dx+ 1 2δ2

Z

Ω

|∇ψ₀|2dx.

(3.5)

Φ being continuous and (u1, u2) bounded, we then obtain:

2

X

i=1 γ_i δi

Z

QT

(∂ui

∂t )

2_dxdt+ 2

X

i=1

1 2δi

Z

Ω

whence (3.3) and (3.4).

Proof of theorem 3 : Differentiating equation ∂bi(ui)

∂t −div[∇ui] +

Now multiplying (3.7) byu′_i, and integrating over Ω gives

1

TheL∞ estimate and hypothesis ( H9) imply successively :

Z by Young’s inequality that :

|ui′|3_L3_(Ω)≤c|u′_i|

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

r′(t) +1

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

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

r(t)≤c(t0), for any ∀t≥t0. (3.15) Now, by (3.15) and hypothesis (H1) , we get :

2

X

i=1

Z

Ω

∂bi(ui) ∂t

2

dx≤M r(t)≤c(t0) f or any t≥t0 .

Then (3.0) is satisfied. Now, as we have :

−△ui=−fi(x, u)−bi(ui)t∈L∞(t0,+∞;L2(Ω)),

then by (S) ui(., t) is in bounded subset of H2(Ω).Hence estimate (3.2)

follows.

For a solution (u1, u2) of (S), we define theω−limit set by :

ω(ϕ₀, ψ₀) =

w= (w1, w2)∈ H1

0(Ω)×L∞(Ω)

∩ H1

0(Ω)×L∞(Ω)

∃tn→+∞ u1(., tn)→w1 inL2(Ω), u2(., tn)→w2 inL2(Ω)

.

Corollary 1. Under the assumptions (H1) to (H10), we haveω(ϕ₀, ψ₀)6=∅

and any (w1, w2) ∈ ω(ϕ₀, ψ₀) is a bounded weak solution of the stationary problem

−∆ui+fi(x, u1, u2) = 0 in Ω

ui= 0 on ∂Ω

Proof : From (3.3) we obtain ω(ϕ₀, ψ₀)6= ∅. Setting wi = Lim

n→∞ ui(., tn) and w= (w1, w2)∈ ω(ϕ₀, ψ₀), we get that w = (w1, w2) is a solution of the Dirichlet problem for elliptic system. The proof is analogous to El Ouardi and de Th´elin [12] and is omitted here.

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

A ⊂ W2,6_(Ω)2

ifN= 3 and

A ⊂ W2,r_(Ω)2

for allr <∞ifN ≤2.

Proof: Taking the inner product of (4.7) withu′′

i,we get d

dt( 2

P

i=1

ku′_ik2

H1₀(Ω))≤c(

2

P

i=1

|u′_i|4

H1₀(Ω)+

2

P

i=1

|u′_i|2

L2(Ω)).

By uniform Gronwall’s lemma, we get

2

P

i=1

ku′_ik2

H1_{0 (Ω)} ≤c,∀t≥T.

Then

2

P

i=1

ku′

ik 2

Lαi(Ω) ≤ c, ∀t ≥ T for all t ≥ τ and αi = 6 if N = 3 or

1≤αi<∞ifN ≤2.

4.1 Linearized problem

Let (ϕ0, ψ0) ∈ A; then by theorem3, u(t) = (u1(t), u2(t)) belongs to a

bounded subset of

H2_(Ω)2

. This fact allows us to linearize the system ( S) alongu(t).Formally, the candidate for the linearized problem is

(SL)

   

  

U = (U1, U2)∈

L2_{(0, T;H}1 0(Ω)

2

∂ ∂t(b

′

i(uiUi)− △Ui+ 2

P

j=1 ∂fi

∂ujUj= 0

U(0) = (U1(0), U2(0)) =U0

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

Ui∈L∞(0, T;L2(Ω))∩L2(0, T;H01(Ω)). (4.0)

To deduce (4.0), Multiply the equation in (SL) byb′i(ui)Ui,we obtain

1 2

d dt(

2

X

i=1

|b′i(ui)Ui| 2 L2_(Ω)) +

2

X

i=1

(∇ui,∇(b′i(ui)Ui))L2_(Ω)

=

2

X

i=1

(

2

X

j=1 ∂fi ∂uj

Uj, b′_i(ui)Ui)_{L2 (Ω)} (4.1)

By the hypothesis onbi, we have

∇(b′_i(ui)Ui) =b′_i(ui)∇Ui+b′′_i(ui)∇ui.Ui, (4.2)

(∇Ui, b′′_i(ui)∇ui.Ui)_L2_(Ω)≤c|∇ui|_L4_(Ω)|Ui|_L4_(Ω)|∇Ui|_L2_(Ω), (4.3)

and

2

X

i=1

(

2

X

j=1 ∂fi ∂uj

Uj, b′_i(ui)Ui)_{L2 (Ω)} ≤M 2

X

i=1 2

X

j=1

|UjUi|

L1(Ω) ≤c

2

X

i=1

|Ui|2

L2(Ω). (4.4)

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

1 2

d dt(

2

X

i=1

|b′_i(ui)Ui|2_L2_(Ω)) +γ

2

X

i=1

|Ui|2

H1_{0 (Ω)} ≤

2 cX

i=1

|Ui|2

L2(Ω) ≤c

2

X

i=1

|b′_i(ui)Ui|2_L2_(Ω) .

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

4.2 Differentiability of the Semigroup

We assume thatfi∈ C2_{(R×R) (∀i= 1,2). Letu0=(ϕ}

0, ψ0), v0=(ϕ0, ψ0), S(t) be the solution of (S) andS′(t, u0) the solution of (SL).The results of [6]

Proposition 3.

Assume (H1) to (H10), then for any (u0, v0)∈

L∞(Ω)×H2(Ω)2

, we have

|S(t)v0−S(t)u0−S′(t, u0)(v0−u0)|

(L2(Ω))2 ≤c(t)o(|v0−u0|(L2(Ω))2)

We need the lemma 3 for the proof of proposition 3:

Let (u1, u2) and (v1, v2) be two solutions of (S) inA with (u1(0), u2(0)) = (ϕ0, ψ0) and (v1(0), v2(0)) =(ϕ1, ψ1). Settingw1=u1−v1 andw2=u2−v2,we

have

Lemme 3Assume (H1) to (H10). For allT >0,there existsc(T)>0 such that for allt∈[0, T],

2

X

i=1

|wi(t)|2_H1

0(Ω)≤c(T)

2

X

i=1

|wi(0)|2_H1

0(Ω), (4.7)

t 2

X

i=1

|wi(t)|2_H1

0(Ω)≤c(T)

2

X

i=1

|wi(0)|2_L2_(Ω), (4.8)

and

2

X

i=1

|w_i′(t)|_L22_(0,T;L2_(Ω))≤c(T)

2

X

i=1

|wi(0)|2H1

0(Ω) (4.9)

Proof : We have

 



∂bi(ui)

∂t − △ui+fi(x, u) = 0 u(0) = (u1(0), u2(0)) = (ϕ0,ψ0)

 



∂bi(vi)

∂t − △vi+fi(x, v) = 0 v(0) = (v1(0), v2(0)) = (ϕ1,ψ1) .

Thus, the differencewi=ui−vi satisfies

b′_i(ui)w′_i− △wi= [b′_i(vi)−b′_i(ui)]v_i′+fi(v)−fi(u). (4.10) Setting

F11=R1

0 ∂f1

∂u1(x, u1+θ(u2−u1), u2)dθ, F21=

R1

0 ∂f1

∂u2(x, u1, u2+θ(u2−u1))dθ,

F12=R1

0 ∂f2

∂u1(x, u1+θ(u2−u1), u2)dθ andF22=

R1

0 ∂f2

∂u2(x, u1, u2+θ(u2−

u1))dθ,

(4.10) becomes

b′_i(ui)w′i− △wi= [bi′(vi)−bi′(ui)]vi′+ 2

X

i=1

We multiply (4.11) by wi

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

(SL)

This problem can be rewritten as

(L)

Let U1, ..., Um be m solutions of (L) corresponding to the initial data

ξ_,, ..., ξ_mandQm(τ) be the orthogonal projector in H =L2(Ω)×L2(Ω) such

k=1is an orthonormal

b′′_i(ui)∈(L∞(0,+∞, L∞(Ω)))2_,so

Now, we have

2

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

2

Which leads to

We setρ(x) =

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

T r(F′(u(τ))◦ Qm(τ)≤ −c

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

( i ) dimF ractale(A)≤ 2m

( ii ) dimH(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 Math´ematiques Appliqu´ees et Industrielles Facult´e des Sciences, B.P 20, El Jadida - Maroc

e-mail adress: aelhachi@yahoo.fr

H. El Ouardi: UFR Math´ematiques Appliqu´ees et Industrielles Facult´e des Sciences, B.P 20, El Jadida - Maroc

and

Ecole Nationale Sup´erieure d’Electricit´e et de M´ecanique B.P 8118-Casablanca-Oasis - Maroc