Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium

www.elsevier.com/locate/anihpc

Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium

Messoud A. Efendiev

, Mitsuharu Ôtani

aHelmholtz Center Munich, Institute of Biomathematics and Biostatistics, Ingolstädter, Landstrasse 1, 85764 Neuherberg, Germany bDepartment of Applied Physics, School of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received 17 November 2010; accepted 21 March 2011 Available online 2 April 2011

Dedicated to Professor Alain Haraux on the occasion of his 60th birthday

Abstract

In this paper we give a detailed study of the global attractors for parabolic equations governed by thep-Laplacian in a heteroge- neous medium. Not only the existence but also the infinite dimensionality of the global attractors is presented by showing that their ε-Kolmogorov entropy behaves as a polynomial of the variable 1/εasεtends to zero, which is not observed for non-degenerate parabolic equations. The upper and lower bounds for the Kolmogorovε-entropy of infinite-dimensional attractors are also obtained.

©2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

In order to describe the long-time behaviour of many dissipative systems generated by the evolution of PDEs in mathematical physics, one often uses the notion of the so-called global attractor, which is a compact invariant set in the phase spaceX which attracts the images of all bounded subsets under the temporal evolution. If the global at- tractor exists, its properties guarantee that the dynamical system reduced to the attractorAcontains all the nontrivial dynamics of the initial system. One of the important questions in this theory is in what sense the dynamics reduced to the attractor are finite or infinite dimensional. It is well known that usually for regular (non-degenerate) dissipa- tive autonomous PDEs in a bounded domainΩ, the Kolmogorovε-entropyH_ε(A, X)of their attractorsAhas the asymptotic property such as:

C₁|Ω|log₂(1/ε)H_ε(A, X)C₂|Ω|log₂(1/ε),

where|Ω|denotes the volume ofΩ andC_i (i=1,2)are some constants independent of|Ω|.

* Corresponding author.

E-mail addresses:messoud.efendiyev@helmholtz-muenchen.de (M.A. Efendiev), otani@waseda.jp (M. Ôtani).

1 Partly supported by the Grant-in-Aid for Scientific Research, #21340032, the Ministry of Education, Culture, Sports, Science and Technology, Japan.

0294-1449/$ – see front matter ©2011 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2011.03.006

In contrast to non-degenerate parabolic PDEs, not so much is known on the long-time behaviour for the degenerate case (see [6,12,8,5]). In particular, the degenerate parabolic equations of the form

(E)

⎧⎪

⎪⎨

⎪⎪

⎩

∂u

∂t =pu(x, t )+u(x, t ), (x, t )∈Ω× [0,∞), u(x, t )=0, (x, t )∈∂Ω× [0,∞), u(x,0)=u₀(x), x∈Ω,

with pu=div(|∇u|^p−2∇u),p >2 are investigated in our previous paper [3], where the following features are revealed:

(a) infinite dimensionality of attractor,

(b) polynomial asymptotics of itsε-Kolmogorov entropy,

(c) difference of the asymptotics of theε-Kolmogorov entropy depending on the choice of the underlying phase spaces,

which one cannot observe in non-degenerate cases.

Remark 1.It is also known that some non-degenerate parabolic equations in unbounded domains may possess infinite- dimensional global attractors. For these cases, however, the asymptotics of their Kolmogorov entropy are always logarithmic (see [4]).

Remark 2. It should be noted that the usual method for obtaining lower bounds of the Kolmogorov entropy (or dimension) of attractors is based on the instability index of hyperbolic equilibria (see [13]), which in turn requires a differentiability of the associated semigroup with respect to the initial data. However, this method may not be applicable for degenerate parabolic equations, since the associated semigroups are usually not differentiable.

The main purpose of the present paper is to give a detailed study of the global attractors for much more wider class of parabolic equations withp-Laplacian in a heterogeneous medium, that is

(E)_p

⎧⎪

⎪⎨

⎪⎪

⎩

∂u

∂t =_pu(x, t )−g

x, u(x, t )

+h(x), (x, t )∈Ω× [0,∞), (E1)

u(x, t )=0, (x, t )∈∂Ω× [0,∞), (E2)

u(x,0)=u0(x), x∈Ω, (E3)

whereΩis a bounded domain inR^N with smooth boundary∂Ω.

The present paper is composed as follows. In Section 1, we first prepare basic results on the existence, uniqueness and regularity of solutions of (E1)–(E3) and under these preparations, the existence of the global attractor for the semigroup generated by (E1)–(E3) will be presented in Section 2. The infinite dimensionality of the global attractors and the asymptotics of their Kolmogorov entropy is given in Section 3. In particular, we show that its Kolmogorov entropy admits polynomial asymptotics which shed light on completely new phenomena (see Remark 1). We here note that one cannot apply the direct approach developed in [3] for (E)p, so to achieve our goal, it requires new ideas. To carry this through, we rely on some comparison results and some special scale transformations which will play a very important role in our arguments. Thus parabolic equations withp-Laplacian of the form (E1) give natural examples of dissipative equations of mathematical physics in heterogeneous media with infinite-dimensional attractors. We especially emphasize that the method developed in this paper has a general nature and can be applied to other classes of degenerate evolution equations.

2. Existence of global solutions and a priori estimates

In this section, we investigate the solvability of the initial–boundary value problem (E)p. To this end, we assume thatg(x, ξ )can be decomposed into two parts, the monotone partg0(x, ξ )and the non-monotone partg1(x, ξ ), i.e., g(x, ξ )=g₀(x, ξ )+g₁(x, ξ )and we further assume thatg₀, g₁satisfy the following conditions:

(a) g₀(x,0)=0,g₀(x, ξ )∈C(Ω×R¹)andg₀(x, ξ )is monotone increasing with respect toξ for allx∈Ω. (b) g₁(x, ξ )∈C(Ω×R¹)andg₁(x, ξ )is a globally Lipschitz function with respect toξ, i.e., there exists a constant

L >0 such that sup

x∈Ω

g₁(x, ξ )−g₁(x, η)L|ξ−η| ∀ξ, η∈R¹. (1)

Then (E)padmits a unique global solution in the following sense.

Theorem 1.Assume that(a)and(b)are satisfied. Then, for anyu₀, h∈L²(Ω), there exists a unique solutionuof (E)psatisfying

u∈C

[0, T];L²(Ω)

, √

t u_t∈L²

0, T;L²(Ω)

, _pu∈L²_loc

(0, T];L²(Ω) , u∈L^p

0, T;W₀^1,p(Ω)

, G0(u)∈L¹(0, T ), t∇u^p_Lp(Ω), tG0(u)∈L^∞(0, T ), ∀T ∈(0,∞), whereG0(u)= _ΩG₀(x, u(x, t )) dx, G₀(x, ξ )= ₀^ξg₀(x, s) ds.

FurthermoreS(t ):u₀(·) →u(·, t )is continuous in the strong topology ofL²(Ω).

Proof. By assumption (a), it is immediate to see thatu → −_pu+g₀(·, u)is monotone inL²(Ω). However, from (a) alone it is impossible to say whether it is maximal monotone inL²(Ω). Hence, to show the existence of solutions for (E)p, we cannot rely on the solvability of abstract evolution equations governed by maximal monotone operators with Lipschitz perturbations (see [1]). To cope with this difficulty, we introduce some approximation procedure and make use ofL^∞-Energy Method (see [10,11]) and the smoothing effect of thep-Laplacian inL^∞-space.

Step 1.Letu₀, h∈L^∞(Ω)and consider

(E)^M_p

⎧⎪

⎨

⎪⎩

∂u

∂t −pu+∂IM(u)+g(x, u)h, (x, t )∈Ω× [0,∞),

u(x, t )=0, (x, t )∈∂Ω× [0,∞),

u(x,0)=u₀(x), x∈Ω,

(2)

whereI_M(·)is the indicator function of the closed convex set{u∈L²(Ω); |u|Ma.e.x∈Ω}:

I_M(u)=0 if|u(x)|Ma.e.x∈Ω, +∞ otherwise

and∂I_M(·)denotes its subdifferential operator given by

∂IM(u)=

0 if|u(x)|< M, [0,+∞] ifu(x)=M, [−∞,0] ifu(x)= −M.

(3) Here we defineϕM(·)by

ϕ_M(u)= ₁

p∇u^p_Lp+I_M(u) if|u(x)|Ma.e.x∈Ω, u∈W₀^1,p(Ω),

+∞ otherwise.

Thenϕ_M(·)is a lower semicontinuous convex functional fromL²(Ω)into[0,+∞]and its subdifferential ∂ϕ_M(·) satisfies

∂ϕM(u)= −pu+∂IM(u), D(ϕM)=

u∈W₀^1,p(Ω); u(x)Ma.e.x∈Ω , D(∂ϕM)=

u∈D(ϕM); pu∈L²(Ω) .

By puttingB(u)=g(·, u(·)), we can reduce our approximate equation (E)^M_p to the following abstract equation:

(AE)^M_p du

dt(t )+∂ϕ_M u(t )

+B u(t )

h, u(0)=u₀.

In order to assure the existence of global solutions of (AE)^M_p, we apply Theorem III and Corollary IV of [9] by taking H =L²(Ω),ϕ^t(·)=ϕ_M(·),B(t,·)=B(·)=g(·, u(·)). To this end, we need to check compactness condition (A1), demiclosedness condition (A2) and boundedness conditions (A5) and (A6) given in [9]. In fact, for anyL >0, the level set{u; ϕ_M(u)+ u²_H L}is compact inL²(Ω)by virtue of Rellich’s compact embedding theorem, which assures (A1). The demiclosedness of the operatorB:u →B(u)=g(·, u(·))inL²(Ω)×L²(Ω)(i.e., the graphG(B) ofBis closed inL²_s(Ω)×L²_w(Ω)endowed with the strong topologyL²_s and the weak topologyL²_w) is easily derived from (a) and (b), whence (A2) follows. As for the boundedness condition, sinceu∈D(∂ϕM)implies that|u(x)|M, we easily get

B(u)

L²(Ω)C_M ∀u∈D(∂ϕ_M),

whence follow (A5) and (A6). Thus we see that for anyu₀∈D(∂ϕ_M)= {u; |u(x)|M}andh∈L²(Ω), (AE)^M_p admits a global solutionusatisfyingu∈C([0, T];L²(Ω)),√

tu_t,√

t _pu∈L²(0, T;L²(Ω))∀T >0. Furthermore, the uniqueness follows easily from the monotonicity ofu → −_pu+g₀(·, u(·))inL²(Ω)and the Lipschitz continuity ofu →g₁(·, u(·))inL²(Ω).

A priori estimate 1.We multiply (E)^M_p by|u|^r−2uand integrate overΩ to get u(t )^r−1

L^r

d dtu(t )

L^r +(r−1)

Ω

|u|^r−2|∇u|^pdx+

Ω

∂I_M(u)|u|^r−2u dx

+

Ω

g0(x, u)|u|^r−2u dx

Ω

g₁(x, u)+h(x)|u|^r−2u dx Lu^rL^r +C

1+ hL^∞

u^r_L^−r1.

Here we used (1) and the fact that∂I_M(u)|u|^r−2u0,g₀(x, u)|u|^r−2u0, which are assured by (3) and (a). Hence, by Gronwall’s inequality, we have

sup

0tT

u(t )

L^r

u0L^r +C

1+ hL^∞

T e^LT. Here lettingr→ ∞, we obtain (see [10,11])

sup

0tT

u(t )

L^∞

u₀L^∞+C

1+ hL^∞

T

e^LT =:C_T. (4)

For eachu₀∈L^∞(Ω)andT >0, fixMso thatC_T < M, then by virtue of (3) and (4), we find that∂I_M(u(t ))=0

∀t∈ [0, T], which implies thatu(t )gives a solution of (E)pon[0, T].

Step 2.Letuⁿ₀, hⁿ∈L^∞(Ω),uⁿ₀→u₀,hⁿ→hinL²(Ω)asn→ ∞, and letuⁿbe the global solution of (E)pwith u₀=uⁿ₀,h=hⁿ, i.e.,

(E)_p,n uⁿ_t −_puⁿ+g₀ x, uⁿ

= −g₁ x, uⁿ

+hⁿ, uⁿ₀(0)=uⁿ₀.

A priori estimate 2.We first note that the monotonicity ofg₀(·, ξ )and the definition ofG₀imply G0(x, ξ )=

ξ 0

g0(x, s) dsg0(x, ξ )ξ.

Hence, multiplying (E)p,nbyuⁿ, we get 1

2 d

dtuⁿ(t )²

L²+∇uⁿ(t )^p

L^p+G0

uⁿ(t )

g₁

·, uⁿ

L²+hⁿ

L²uⁿ(t )

L²

Luⁿ(t )²

L²+

C+hⁿ

L²uⁿ(t )

L², (5)

whence it follows that sup

0tT

uⁿ(t )²

L²+ T

0

∇uⁿ(t )^p

L^p+G0

uⁿ(t )

dtC₀

u₀L²,hL²

. (6)

Next multiply (E)p,nbyt (uⁿ)_t and integrate over[0, T]with respect tot, then we have tuⁿ

t²

L²+t d dt

1

p∇uⁿ(t )^p

L^p+G0

uⁿ(t )

g₁

uⁿ(t )

L²+hⁿ

L²

tuⁿ

t

L² 1 2tuⁿ

t²

L²+C₀, sup

0tT

t 1

p∇uⁿ(t )^p

L^p+G0

uⁿ(t ) +

T 0

tuⁿ

t²

L²dtC₀. (7)

Hence, from Eq. (E)p,n, we derive the a priori bound √

t

−_puⁿ+g₀ x, uⁿ

_L²_(Q)C₀, Q=Ω×(0, T ). (8)

However, this does not assure the boundedness of√

t _puⁿL²(Q). Nevertheless, by applying a comparison theorem, which will be given in Lemma 1, betweenuⁿ andv^±, solutions of (17) satisfying (20), we can derive some a priori estimate for uⁿ in L^∞(Ω) as follows. We first note that v^± also satisfy (15), so by the same reasoning as used for (12), we can obtain the a priori bound forv^±_L²_(Ω). Then integrating (15) on[0, T], we derive, by Poincaré’s inequality, the a priori bound forv^±L^p(Q)depending only onu₀L²(Ω),hL²(Ω). Hence the right-hand side of (20) is bounded by some constant independent ofnon [δ, T]for eachδ >0, which together with (18) yields the following estimate:

∀δ >0,∃Cδ=C(δ, C0) s.t. sup

δtT

uⁿ(·, t )

L^∞(Ω)Cδ, whence, by (8), it follows that

sup

δtT

g₀

·, uⁿ(t )

L∞(Ω)C_δ, _puⁿ

L²(δ,T;L²(Ω))C_δ. (9)

Convergence.w=u^m−uⁿsatisfies w_t−

_pu^m−_puⁿ +

g₀ x, u^m

−g₀ x, uⁿ

= − g₁

x, u^m

−g₁ x, uⁿ

. (10)

Multiplying (10) bywand using the monotonicity ofu → −_pu+g₀(·, u)inL²(Ω)and condition (b), we get 1

2 d

dtw(t )²

L² Lw(t )²

L², whence it follows that

w(t )

L²u^m₀ −uⁿ₀

L²e^Lt,

which implies that{uⁿ}forms a Cauchy sequence inL²(Ω). Thus, in view of (6), (7) and (9), we find that there exists a subsequence of{uⁿ}denoted again by{uⁿ}such that

uⁿ→u strongly inC

[0, T];L²(Ω)

, and a.e.(x, t )∈Q, g₁

x, uⁿ

→g₁(x, u) strongly inC

[0, T];L²(Ω) ,

∇uⁿ→ ∇u weakly inL^p

[0, T];L^p(Ω)

√ ,

t uⁿ_t →√

t u_t weakly inL²(Q), g0

x, uⁿ

→g0(x, u) weakly star inL^∞(Qδ),

_puⁿ→_pu weakly inL²(Q_δ), Q_δ=Ω× [δ, T], ∀δ >0.

Here we used the demiclosedness of g₀(·, u),−_pu, i.e., their graphs are closed inHs×Hw, whereHs andHw

denoteL²(Q_δ)endowed with the strong topology and the weak topology respectively. Furthermore, by virtue of the lower-semicontinuity of∇u^p_L^p,G0(u), we easily see thatt|∇u|^pL^∞(0,T;L¹(Ω)),G0(u)L¹(0,T ),tG0(u)L^∞(0,T )

are all bounded. 2

Remark 3.Ifg(x, u)∈C(Ω×R¹)satisfiesg(x,0)=0,g_τ(x, τ )−K∀(x, τ )∈Ω×R¹,then by puttingg₀(x, τ )= g(x, τ )+Kτ ((g₀)_τ(x, τ )0),g₁(x, τ )= −Kτ,we find thatg(·, u)falls within our class.

In particular, g(x, τ )=C₁(x)|τ|^q1−2τ−C₂(x)|τ|^q2−2τ (2q₂< q₁<+∞)satisfies (a) and (b), provided that C₁(x), C₂(x)∈L^∞(Ω)and 0< c₁C₁(x)∀x∈Ω. Sinceg_τ(x, τ )= |τ|^q2−1(C₁(x)(q₁−1)|τ|^q1−q2−C₂(x)(q₂− 1))−K∀(x, τ )∈Ω×R¹.

As for the a priori bounds for solutions of (E1)–(E3), we obtain the following result.

Theorem 2.Assume that(a)and(b)are satisfied and let _N^2N₊₂< p <∞, u₀∈L²(Ω)andh∈L^∞(Ω). For the case

2N

N+2< p2, we further assume

g₀(s)k₀|s|^1+θ−k₁ (θ, k₀, k₁>0). (11)

Then, for any0< δ₁< δ₂< δ₃1, there exist constantsC₁,C₂,C₃depending onδ₁,δ₂,δ₃but not on the initial data u₀∈L²(Ω)such that every solution of (E1)–(E3)satisfies

u(t )

L²(Ω)C1 ∀t∈ [δ1,+∞), (12)

u(t )

L^∞(Ω)C2 ∀t∈ [δ2,+∞), (13)

u(t )

C^1,α(Ω)C3 ∀t∈ [δ3,+∞). (14)

Proof. Multiply (E1) byu, then the same argument as for (5) gives 1

2 d

dtu(t )²

L²+ ∇u^p_L^p+G0(u)Lu(t )²

L²+

C+ hL²

uL². (15) Then by Poincaré’s inequality, we get for the casep >2

1 2

d

dtu(t )²

L²+γ₁u^p_L2γ₂, (16)

for someγ₁, γ₂>0. Hence Ghidaglia-type estimate (see [13]) assures (12). For the case_N^2N₊₂p2, (11) yields the same estimate as (16) withp=2+θ, whence follows (12).

In order to derive theL^∞-estimate, we need the following lemma.

Lemma 1.Letv^±be the unique solution of v^±_t =pv^±−k0v^±θv^±−g1

v^±

±

|h| +k1

, v^±

∂Ω=0 (17)

with the initial conditionv^±(x,0)= ±|u₀(x)|respectively, wherek_i =0forp >2andk_i=k_i forp2(i=0,1).

Then the solution u of (E)psatisfies

v⁻(x, t )u(x, t )v⁺(x, t ) for a.e.(x, t )∈Ω× [0,+∞). (18) Proof. Letu⁺(x, t )be the unique solution of (E)pwithhandu₀replaced by|h|and|u₀|respectively.

Then it is easy to see thatu⁺0 and(u⁺−v⁺)satisfies u⁺−v⁺

t=_pu⁺−_pv⁺−g₀ u⁺

+k₀v^+θv⁺−k₁−g₁ u⁺

+g₁ v⁺

. (19)

Multiply (19) by[u⁺−v⁺]⁺=max(u⁺(x, t )−v⁺(x, t ),0). Then noting that

_pu⁺−_pv⁺,

u⁺−v⁺₊

= −∇

u⁺−v⁺₊^p

L^p0, −g₀

u⁺

+k₀v^+θv⁺−k₁,

u⁺−v⁺₊

−k₀u^+θu⁺−v^+θv⁺,

u⁺−v⁺₊

0,

−g₁ u⁺

+g₁ v⁺

,

u⁺−v⁺₊

Lu⁺−v⁺₊²

L², we get

u⁺−v⁺₊(t )²

L²u⁺−v⁺₊(0)²

L²e^2Lt,

whence follows 0u⁺v⁺. By much the same argument as above, we also find uu⁺. Thus repeating this procedure foru⁻(the solution of (E)pwithhandu₀replaced by−|h|and−|u₀|) andv⁻, we obtain (18). 2 Proof of Theorem 2 (continued). Since Eqs. (17) have the simple form, theL^∞-estimate forv^± has been fully investigated by many authors. For instance, Theorem 3.2 of Chapter 5 of [2] gives the estimate:

sup

x∈Ω

v^±(x, t )max

1, C

t^Np +1 t

_q¹_−δt 0

Ω

v^±δdx dτ ^p

N (q−δ)

(20) whereδ=2+θforp2,δ=pforp >2 andq=p(N+2)/N.

Here [2] requires the condition pδ < p^N_N⁺², which is obviously satisfied forp >2. As for the case where

2N

N+2< p2, since _N^2N₊₂< pimplies 2< p^N_N⁺², we can choose a sufficiently smallθ₀such thatp2+θ₀< p^N_N⁺². Ifθ0< θ, it is clear that (11) is satisfied withθandk1replaced byθ0andk0+k1respectively. As is seen in the proof of Theorem 1, a priori estimate 2,v^±satisfy (16) withp=porp=2+θ. Hence, integrating this on[t, t+1], we get, by (12)

sup

t1

v^±

L^δ(t,t+1;L^δ(Ω))C1. (21)

Thus the estimate (13) is derived from (18), (20) and (21).

Now we can rewrite (E1) asu_t(x, t )=_pu(x, t )+h(x, t ),whereh(x, t )= −g₀(x, u(x, t ))−g₁(x, u(x, t ))+ h(x). Note that (13) assuresh∈L^∞(Ω× [δ2,+∞)). Consequently, by virtue of Theorem 1.2 in Chapter 10 of [2], we can derive theC^1,α(Ω)-bound foruon[δ3,+∞). 2

3. Global attractor and Kolmogorov’sε-entropy

LetΦ be a Banach space. The setA⊂Φ is called a global attractor of the semigroupS(t )inΦ if the following conditions are satisfied:

(1) The setAis a compact subset of the phase spaceΦ. (2) It is strictly invariant, i.e.,S(t )A=Afor allt0.

(3) For every bounded subsetB⊂Φ, limt→∞dist(S(t )B,A)=0,where dist(X, Y )=sup_x∈Xinfy∈Yx−yΦ. Our existence result for global attractors of (E1)–(E3) inΦ=L²(Ω)can be stated as follows.

Theorem 3.Let all the assumptions in Theorem2be satisfied. Then the semigroupS(t )associated with Eqs.(E1)–

(E3)possesses a global attractorAin the phase spaceL²(Ω)which is globally bounded inC^1,α(Ω)withα∈(0,1]

and has the following structure:A:=K0:= {u(0); {u(t )}_t∈R¹ ∈K},whereKis the set of all bounded solutions of (E1)–(E2)defined onR¹, i.e.,

K= u(t )

t∈R¹; u(t )is a solution of (E1)–(E2)onR¹, sup

t∈R¹

u(t )

L²<+∞

.

Proof. In order to prove the existence of the global attractorAfor (E1)–(E3), it suffices to show that the semigroup S(t )associated with (E1)–(E3) is continuous in the topology ofL²(Ω)for eacht >0 and that there exists a pre- compact absorbing setBinL²(Ω)such that for everyx∈L²(Ω), there existsT =T (x) >0 such thatS(t )x∈B

∀t∈ [T ,+∞). (See [13, Theorem 1.1].) For our case, the first property is assured by Theorem 1 and the second by Theorem 2. The characterization ofAin terms ofKis derived by standard arguments. 2

Next we present lower bounds for Kolmogorov’sε-entropy of the attractorAinΦ=L^p(Ω) (1p∞), denoted byHε(A, Φ), which is defined by the base 2 logarithm ofNε(A, Φ), that is,Hε(A, Φ):=log₂Nε(A, Φ). Here we denote byNε(A, Φ)the minimal number ofε-balls inΦ that coversA(recall thatAis a compact set inΦ). From now on we assume thatp >2,h≡0 andg(·, u)satisfies the following assumption.

(I)g There exist an open bounded subsetωofΩ andα >0 such thatg^α(x, u)=g(x, u)+αusatisfies:

(I)1 There exista(s),ρ0 andh(x, s, v)satisfying s⁻

p+1 p−2 g^α

x, s^p1−2v

=a(s)|v|^ρh(x, s, v), (x, s, v)∈ω× [0,1] ×R¹, h(x, s, v), h_v(x, s, v)∈C

ω× [0,1] ×R¹

, h(x, s,0)=0, a0, a∈L¹(0,1), sa²∈L¹(0,1).

(I)2 There existC >0 andδ >0 such that

|v|^ρh(x, s, v)C|v|^1+δ, (x, s, v)∈ω× [0,1] × [0,1].

Then our result on the infinite dimensionality of global attractors for (E1)–(E3) is as follows.

Theorem 4.Let(I)gbe satisfied and assume that(E1)–(E3)possess a global attractorAin the topology ofL²(Ω).

Then the fractal dimension ofAis infinite.

The presentation of condition (I)gmight be somewhat obscure. In order to clarify the meaning of this condition, we show below that it covers a very large class of nonlinearity.

(Ex.1).Letg(x, u)= −αu+b₁(x)|u|^q1−2u−b₂(x)|u|^q2−2u,α >0, 2< q₂< q₁,b₁, b₂∈C(ω),theng(·, u)satis- fies (I)g.

In fact, sinceg^α(x, u)=b₁(x)|u|^q1−2u−b₂(x)|u|^q2−2u,we get s⁻

p+1 p−2 g^α

x, s^p−12v

=s⁻

p+1 p−2

b₁(x)s

q1−1

p−2|v|^q1−2v−b₂(x)s

q2−1

p−2|v|^q2−2v

=s

q2−p

p−2 |v|^q2−2 b₁(x)s

q1−q2

p−2 |v|^q1−q2v−b₂(x)v . Hence we can put

a(s)=s

q2−p

p−2 , ρ=q2−2>0, h(x, s, v)=b1(x)s

q1−q2

p−2 |v|^q1−q2v−b2(x)v.

Then it is easy to see that h(x, s, v), h_v(x, s, v)∈C

Ω× [0,1] ×R¹

, h(x, s,0)=0,

|a|_L¹= 1 0

s

q2−p

p−2 ds= p−2

q2−2, sa²

L¹= 1 0

ss

2q2−2p

p−2 ds= p−2 2q2−4,

|v|^ρh(x, s, v)C|v|^q2−1=C|v|^1+δ, (x, s, v)∈ω× [0,1] × [0,1], δ=q₂−2>0.

(Ex.2).Letg(x, u), g_u(x, u), g_u(x, u)∈C(ω×R¹),g_u(x,0)= −α <0,g(x,0)=0,theng(·, u)satisfies (I)g. In fact, we first note that

g(x, u)=g(x,0)+g_u(x,0)u+ u 0

(u−t )g_u(x, t ) dt, g^α(x, u)= u 0

(u−t )g_u(x, t ) dt.

Then we get

s⁻

p+1 p−2 g^α

x, s^p−12v

=s⁻

p+1 p−2

s

1 p−2v

0

s^p−12v−t

g_v(x, t ) dt

=s⁻

p+1 p−2

v 0

s^p−21 v−s^p−21 t g_v

x, s^p−21 t s^p−21 dt

=s

3−p p−2

v 0

(v−t )g_v

x, s^p−12t dt,

and we put

ρ=0, h(x, s, v)= v 0

(v−t )g_v

x, s^p−21 t

dt, a(s)=s

3−p p−2.

Hence we obtain

h(x, s, v), h_v(x, s, v)= v 0

g

x, s^p1−2t dt∈C

ω× [0,1] ×R¹

, h(x, s,0)=0,

|a|L¹ = 1 0

s

3−p

p−2ds=(p−2), sa²

L¹= 1 0

ss

6−2p

p−2 ds=p−2 2 ,

|v|^ρh(x, s, v)= v 0

(v−t )g_v

x, s^p−21 t dt

|v|

0

1−t v

g_v

x, s^p−21 tdt|v|

max

(x,s)∈ω×[0,1]g(x, s)|v|², (x, s, v)∈ω× [0,1] × [0,1], δ=1.

In order to establish the estimate from below forε-Kolmogorov entropy of our global attractorA, we rely on the following fact.

Lemma 2.LetK⁻be the set of all bounded solutions of(E1)–(E2)onR¹₋, i.e.,K⁻= {{u(t )}_t∈R¹

−; u(t )satisfies(E1)–

(E2)onR¹₋,sup_t∈R1

−u(t )_L²<+∞}, whereR¹₋:=(−∞,0]and letK⁻(t )be the section ofK⁻att=t∈R¹₋, i.e., K⁻(t )= {u(t ); {u(t )}_t∈R¹₋∈K⁻}. ThenK⁻(0)⊂Aholds true.

Proof. PutB=

t∈R¹₋K⁻(t ). Then, sinceBis bounded inL²(Ω), for arbitraryη >0, there existsT >0 such that dist(S(T )B,A) < η. For anya0∈K⁻(0), there existsaT ∈K⁻(−T )⊂Bsuch thatS(T )aT =a0. Hence we get

dist(a0,A)=dist

S(T )a_T,A dist

S(T )B,A

< η, ∀η >0,

which implies dist(a0,A)=0, i.e.,a₀∈ ¯A=A∀a₀∈K⁻(0). ThusK⁻(0)⊂Ais derived. 2

Before we proceed to the proof of Theorem 4, we prepare a couple of results on the following auxiliary equation:

(E)^t_p

w_t=p_α

_pw−a(t )|w|^ρh(x, t, w)

, (x, t )∈ω×(0,1),

w|∂ω=0, t∈ [0,1], w(x,0)=w₀(x), x∈ω, (22) wherep_α=_α(p¹₋₂₎. As for the solvability of this equation, the following result holds.

Lemma 3.Let(I)gbe satisfied. Then for everyw₀∈L^∞(ω), there existsT₀=T₀(w₀L^∞) >0such that(E)^t_padmits a unique solutionwon[0, T0]satisfying

w∈C

[0, T0];L²(ω)

∩C

(0, T0];W₀^1,p(ω)

∩L^∞

ω×(0, T0)

√ , t wt,√

t pw∈L²

ω×(0, T0)

. (23)

Furthermore there exists a(sufficiently small)ε₀>0such that ifw₀L^∞ε₀, then the solutionw of(E)^t_p given above can be continued up to[0,1]and satisfiessup_t∈[0,1]w(t )L^∞1.

Proof. We again apply theL^∞-Energy Method as in the proof of Theorem 1 (Step 1). Take 0an(t )∈C([0,1])so thatan(t )→a(t )inL¹(0, T )and√

t an(t )→√

t a(t )inL²(0, T )asn→ ∞. PutM= w0L^∞+1 and consider the following equations:

(E)^t,M_p,n

wⁿ_t ∈pα

pwⁿ−∂IM

wⁿ

−an(t )w^nρh

x, t, wⁿ

inω×(0,1), wⁿ

∂ω=0, t∈ [0,1], wⁿ(x,0)=w₀(x), x∈ω, (24)

where ∂I_M(·)is the subdifferential of I_M(·), given by (3). As in the proof of Theorem 1, we can reduce (E)^t,Mp,n

to some abstract Cauchy problem such as (AE)^M_p, and apply Theorem III and Corollary IV of [9] by takingH= L²(Ω),ϕ^t(·)=p_αϕ_M(·),B(t, w)=p_αa_n(t )|w|^ρh(x, t, w). Here we note thatw₀∈D(∂ϕ_M)= {u; |u(x)|M= w₀L^∞+1}. Thus the existence of solutions of (24) is verified.

A priori estimates.We first note that h(x, s, v)=

v 0

h_τ(x, s, τ ) dτ 0

|v|

|v|,

0(r)=maxh_v(x, s, v); x∈ω, s∈ [0,1], |v|r .

Here ₀(r) is monotone increasing inr∈ [0,+∞). Then, as in the proof of Theorem 1 (a priori estimate 1), by multiplying (E)^t,Mp,n by|u|^r−2u, we get

1 r

d

dtwⁿ(t )^r

L^r pαan(t )wⁿ(t )^ρ

L^∞0wⁿ(t )

L^∞wⁿ(t )^r

L^r, wⁿ(t )

L^r w₀L^r+ t 0

a_n(s)₁wⁿ(s)

L^∞wⁿ(s)

L^rds, with₁(r)=p_αr^ρ₀(r).

Then lettingr→ ∞, we obtain (see [10,11]) wⁿ(t )

L^∞w₀L^∞+ t 0

a_n(s)wⁿ(s)

L^∞

ds, with(r)=₁(r)r. (25)

Here define a positive numberδand chooseT₀such that

δ= 1

(M)+1,

T₀

0

a(s) ds < δ

2. (26)

Then, sincea_n→ainL¹(0,1), there existsNsuch that

T₀

0

a_n(s) ds < δ ∀nN. (27)