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Optimization: A Journal of

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Long-time behaviour for a model of porous-medium equations with variable coefficients

Tran Dinh Ke ^a & Ngai-Ching Wong ^b

a Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

b Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

Version of record first published: 13 Aug 2010.

To cite this article: Tran Dinh Ke & Ngai-Ching Wong (2011): Long-time behaviour for a model of porous-medium equations with variable coefficients, Optimization: A Journal of Mathematical Programming and Operations Research, 60:6, 709-724

To link to this article: http://dx.doi.org/10.1080/02331934.2010.505963

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Vol. 60, No. 6, June 2011, 709–724

Long-time behaviour for a model of porous-medium equations with variable coefficients

Tran Dinh Ke^aand Ngai-Ching Wong^b*

aDepartment of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam;^bDepartment of Applied Mathematics, National Sun Yat-sen

University, Kaohsiung 804, Taiwan

(Received 10 November 2009; final version received 19 April 2010) By analysing the uniform attractor for multi-valued processes, we study the long-time behaviour of the solutions of a model of non-autonomous porous-medium equations. The result is obtained by using the a priori estimates and suitable compactness arguments.

Keywords: porous-medium equations; degenerate parabolic equations;

non-autonomous; uniform attractors

AMS Subject Classifications: 35B40; 35B41; 35B45; 35D05

1. Introduction

Letbe a bounded domain inR^N(N52) with a smooth boundary@. We consider the following problem

@u

@tdivððxÞr’ðuÞÞ þfðt,uÞ ¼gðx,tÞ, x2, t4, ð1:1Þ

uj_t¼ ¼uðxÞ, x2, ð1:2Þ

uj_@¼0, ð1:3Þ

where2R, and the functions,’,fandgsatisfy some conditions specified later.

The equation of type (1.1) represents the motion of gas through a porous medium whereu(x,t) stands for the gas density. The original model was established by the mass conservation law as follows. LetV be the velocity. Then

utþdivðuVÞ ¼fðuÞ, ð1:4Þ

where f(u) models the effects of reaction or absorption. In the case of a non-homogeneous medium, V ¼ (x)rP(u). Here, P is the pressure and is a given function.

*Corresponding author. Email: wong@math.nsysu.edu.tw

ISSN 0233–1934 print/ISSN 1029–4945 online ß2011 Taylor & Francis

DOI: 10.1080/02331934.2010.505963 http://www.informaworld.com

Downloaded by [National Sun Yat-Sen University] at 06:35 15 December 2012

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There are literature works with different views on this type of equation. In many works, it is usually assumed that P(u)¼u^m with and mas constants and then Equation (1.4) assumes the form

ut m

mþ1divððxÞru^1þmÞ ¼fðuÞ:

For a systematic investigation in the case of¼1, we refer the readers to [23]. In addition, for the problem withf¼0, there are several studies in the situation when m! þ1, in relation to the Hele–Shaw problem. See [3–5,10,12] and related works.

Recently, many works have been carried out for the problem with variants of nonlinearityf. Let us introduce some relevant studies in [7,15,19], among others. For more generalized equations, when m is a variable exponent, see [2,20]. Using an approach similar to that in [1], we study, in this article problem (1.1)–(1.3) when the derivative’⁰is a generalization of homogeneous function,may have zeros at some points inand the nonlinearity offhas the form of a power without upper bound.

Precisely, we assume that

(H1) The function ’2C¹(R) and satisfies mjuj^p24’⁰(u)4Mjuj^p2, where m,M40 and p42.

(H2) 2L¹_locðÞsuch that lim infx!zjxzj(x)40 for some2(0, 2) and for everyz2.

(H3) f:RR!R is a continuous function and jf(t,u)j4Cf(juj^q1þ1) for Cf40,q5pand for all t2R.

(H4) There existsMf40 such thatf(u)u5Mf(juj^q1).

For the external forceg, we impose a restriction that (H5) The functiong2L²_locðR;L²ðÞÞand satisfies

kgk²_L2 b

:¼sup

t2R

Ztþ1 t

kgð,sÞk²_L2ðÞds5þ1:

The class of functions satisfying (H1) includes the functions such that their derivatives are homogeneous of order p2, ’⁰(u)¼ juj^p2 and some generalized forms, such as’⁰(u)¼ juj^p2(u) where(u) is bounded,m4(u)4Mfor allu2R. Hypothesis (H2) is motivated by the work [6] in which a semi-linear elliptic problem was studied. It ensures that has at most finite zeros in . In various processes,(x) jxj. It is worth noting that the degeneracy made byprevents us from using the regularization method as in [2] to get the existence result. Moreover, since q has no upper bounds, we may have a set of solutions to (1.1)–(1.3) corresponding to each initial datum from the phase space. That makes our problem, in general, generating a multi-valued process (MVP). The aim of our work is to prove the existence result for (1.1)–(1.3) and study the asymptotic behaviour of solutions over a large amount of time. To this end, we employ the notion of uniform attractors for MVPs, with respect to the symbol (f,g). For the analysis details, see [13,17,18]

and a similar approach in [8]. This framework can be seen as an extension for the theory of global attractors which was developed in [9,11,21].

In order to study the problem (1.1)–(1.3), we introduce some weighted Sobolev spaces. ByD^1,2₀ ð,Þwe denote the closure ofC₀¹ðÞwith respect to the norm

kvk_D^1,2

0 ð,Þ¼

Z

ðxÞjrvj² ¹₂

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and letVbe the space of all functionsw2L^p() satisfying w@¼0 and @

@xi

@w

@xi

2L^pðÞ; i¼1,. . ., ,N, ð1:5Þ equipped with the norm

kwk_V¼ Z

divðrwÞ^p¹_p

: ð1:6Þ

Putting

aðu,Þ ¼ Z

ðxÞr’ðuÞr, for2V, ð1:7Þ one can write

aðu,Þ ¼ Z

’ðuÞdivðrÞ, ð1:8Þ

aðu,uÞ ¼ Z

ðxÞ’⁰ðuÞjruj²: ð1:9Þ One observes from (H1) that j’(u)j4juj^p1þ j’(0)j, and a(u,) is well-defined if u2L^p() and2V.

Denoting

ðuÞ ¼ Zu

0

ffiffiffiffiffiffiffiffiffiffi

’⁰ðsÞ

p ds ð1:10Þ

for u2R, it is obvious that is an increasing function. Furthermore, it follows from (H1) that

ffiffiffiffim

p juj^p²4jðuÞj4 ffiffiffiffiffi M p

juj^p²: ð1:11Þ

From (1.9) and (1.10), we have aðu,uÞ ¼

Z

ðxÞjrðuÞj²: ð1:12Þ Definition 1.1 We say that a function u(x,t) is a weak solution of (1.1)–(1.3) in Q,T¼[,T] if and only if

u2L^qðQ,TÞ \Cð½,T;L²ðÞÞ, ðuÞ 2 D^1,2₀ ð,Þ,

@u

@t2L^p⁰ð,T;V⁰Þ þL^q⁰ðQ_,TÞ, uj_t¼¼u a.e. in,

and

ZT

aðu,Þdt¼ Z

Q,T

gðx,tÞ fðt,uÞ @u

@t

dxdt ð1:13Þ for every test functions2L^p(,T;V)\L^q(Q,T).

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Equation (1.1) can be seen as an equation in L^p⁰(,T;V⁰)þL^q⁰(Q,T). HereV⁰is the dual space ofV;p⁰andq⁰are the conjugate exponents ofpandq, respectively. In this definition, we assume that ðuÞ 2 D^1,2₀ ð,Þ since we need the definiteness of a(u,u), which is important for our arguments in Section 2.

We first show some compactness results for our purposes in Section 2. In the last section, we state and prove the main results.

2. Preliminaries

We recall the results of Caldiroli and Musina [6] related to the spaceD^1,2₀ ð,Þ.

PROPOSITION 2.1 Assume that is a bounded domain inR^N,N52,and satisfies (H2).Then the following embeddings hold:

(i) D^1,2₀ ð,Þ,!L²ðÞcontinuously,

(ii) D^1,2₀ ð,Þ,!L^rðÞcompactly if r2 ½1, 2Þ, where2¼_N2þ^2N .

By Proposition 2.1, kk_Vin (1.6) is the well-defined norm. Indeed, it suffices to check that ifkwkV¼0 thenw¼0. Forw2V, we have

Z

wdivðrwÞ ¼ Z

jrwj² ¼ kwk²_D1,2

0 ð,Þ5Ce1kwk²_L2ðÞ: In addition

Z

wdivðrwÞ4kwk_L²_ðÞkdivðrwÞk_L²_ðÞ 4Ce₂kwk_L2ðÞkdivðrwÞk_LpðÞ

for someCe₁,Ce₂40. Thus

kwk_V5Ce1

Ce₂kwk_L2ðÞ: The following is the important tools for our arguments.

PROPOSITION 2.2 Let {vn}be a sequence such that a(vn,vn)4C,for some C40,and for all n2N.Then{vn}is precompact in L^p().

Proof From (1.12), we have

aðv_n,v_nÞ ¼ Z

ðxÞjrðv_nÞj²:

Then we observe that the sequence {(vn)} is bounded inD^1,2₀ ð,Þ. By Proposition 2.1, {(vn)} is precompact inL () for all satisfying 15 52and then(vn)! strongly in L (), by replacing with a subsequence if necessary. This ensures that (vn)!a.e. in. In addition, by the boundedness of {(vn)} inL () and (1.11), we see that {v_n} is bounded in L^p²ðÞ. It follows that vn*v in L^p²ðÞ. By the monotonicity of, we deduce that vn!¹()¼va.e. in.

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For any 40, by Egorov’s theorem, there exists a subset E with measure jEj5, such thatvn!vuniformly in\E. Taking r5^p₂, we have

Z

jvnvj^r¼ Z

nE

jvnvj^rþ Z

E

jvnvj^r

4 Z

nE

jvnvj^rþ Z

E

jvnvj^p²

¹, with¼2r p :

The last inequality shows that vn!v strongly in L^r(). Choosing such that 25 52 and then takingr¼p, we get the conclusion. g The next two propositions can be proved by using the arguments as in [16, Section 12, Chapter 1] with some slight modifications. In what follows, we denote by V¼V⁰þL^q⁰().

PROPOSITION 2.3 For any40,there exists C40such that kuvk^p_LpðÞ4½aðu,uÞ þaðv,vÞ þCkuvk^p_V

for all u,v2S¼ fwðwÞ 2 D^1,2₀ ð,Þg.

Proof Assume on contrary that there exist 040 and two sequences un,vn2S such that

kunvnk^p_LpðÞ40½aðun,unÞ þaðvn,vnÞ þnkunvnk^p_V: Putting

eu_n ¼ u_n

½aðu_n,u_nÞ þaðv_n,v_nÞ¹^p

, ev_n¼ v_n

½aðu_n,u_nÞ þaðv_n,v_nÞ¹^p , we have

keu_nev_nk^p_LpðÞ40þnkeu_nev_nk^p_V: ð2:1Þ Noting that

aðtu,tuÞ ¼ jtj² Z

ðxÞ’⁰ðtuÞjruj²4Mjtj^p Z

ðxÞjuj^p2jruj² 4M

mjtj^paðu,uÞ for allt2R, we get

aðeun,eunÞ4M

m, aðevn,evnÞ4M m:

Therefore, by Proposition 2.2, there exist two subsequenceseunm andevnm such that eu_n_m!eu, ev_n_m!ev strongly inL^pðÞ:

Since p42 and VL^p(), it follows that L^p()L^p⁰()V⁰. Moreover, since q5p, one has L^p()L^p⁰()L^q⁰(). This implies that L^p()V. By (2.1) we have keuevk_V¼0, and thereforeeu¼ev. Then keuevk_LpðÞ ¼0, which

contradicts (2.1). g

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PROPOSITION 2.4 Assume that RT

aðunðtÞ,unðtÞÞdt4C and _du_n

dt is bounded in L^p⁰(,T;V⁰)þL^q⁰(Q,T).Then{un}is precompact in L^p(Q,T).

Proof From Proposition 2.3, we have Z_T

kunþ‘unk^p_LpðÞ4 Z_T

½aðunþ‘,unþ‘Þ þaðun,unÞ þC

Z_T

kunþ‘unk^p_V: Hence it suffices to show that the sequence {u_n} contains a subsequence, which is a Cauchy sequence inL^p(,T;V).

We will prove a stronger claim: There exists a subsequenceusuch that u!u in Cð½,T;VÞ: ð2:2Þ By the hypotheses, there exists a set Z[,T] with measurejZj ¼0 such that, for t2[,T]\Z,

aðunðtÞ,unðtÞÞ4Kt51:

Hence for anyt2=Z, there exists a subsequence (dependent oft) such that

u_kðtÞ !uðtÞ strongly inV: ð2:3Þ Now let {t1,t2,. . .} be a sequence which is dense in [,T], andti2=Z. Using (2.3) and a diagonal procedure, we can extract a subsequenceusuch that

uðt_iÞ !uðt_iÞstrongly in V for alli: ð2:4Þ Noting thatL^p⁰(,T;V⁰)þL^q⁰(Q,T)L^q⁰(,T;V), for all t2[,T], we have

kuðt_iÞ uðtÞk_V ¼

Zti

t

u⁰ðsÞds V

4 Zti

t

ku⁰ðsÞk^q_V⁰ds _q¹0

jt_itj¹^q4Cjt_itj¹^q: Then

kuþrðtÞ uðtÞk_V4kuþrðtÞ uþrðtiÞk_V

þ ku_þrðt_iÞ uðt_iÞk_Vþ kuðt_iÞ uðtÞk_V, 4ku_þrðt_iÞ uðt_iÞk_Vþ2Cjt_itj¹^q

for all t2[,T]. By the density of {ti} in [,T], it follows that {u} is a Cauchy sequence inVuniformly int2[,T]. The proof is complete. g We use the next proposition, which makes the initial condition in problem (1.1)–(1.3) meaningful.

PROPOSITION 2.5 If u2L^p(,T;V)\L^q(Q,T)anddu

dt 2L^p⁰ð,T;V⁰Þ þL^q⁰ðQ,TÞthen u2C([,T];L²()).

For the proof, we refer the readers to [14].

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3. Main results

3.1. Existence of a global solution

THEOREM 3.1 Under the assumptions(H1)–(H5), the problem(1.1)–(1.3)has at least one weak solution for each u2L²().

Proof Consider the approximating solutionun(t) in the form u_nðtÞ ¼Xⁿ

k¼1

u_nkðtÞe_k,

wherefejg¹_j¼1is a basis ofD^1,2₀ ð,Þ \L^pðÞ, which is orthogonal inL²(). We getun

from solving the problem dun

dt ,ek

¼ hAun,eki hfðunÞ,eki þ hg,eki, ðu_nðÞ,e_kÞ ¼ ðu,e_kÞ,

whereAu¼div((x)’⁰(u)ru).

Sincef,’⁰2C(R), the Peano theorem ensures the local existence of un. We now establish somea prioriestimates forun. We have

1 2

d

dtku_nk²_L2ðÞþaðu_n,u_nÞ þ Z

fðu_nÞu_n¼ Z

gu_n: Using hypothesis (H4) and the Cauchy inequality, we get

1 2

d

dtkunk²_L2ðÞþaðun,unÞ þMf

Z

junj^q 4Mfjj þ1

2kgk²_L2ðÞþ1

2kunk²_L2ðÞ: ð3:1Þ It follows that

d

dtkunk²_L2ðÞ4kunk²_L2ðÞþ kgk²_L2ðÞþ2Mfjj:

Then

d

dtðe^tkunk²_L2ðÞÞ4e^tðkgk²_L2ðÞþ2MfjjÞ:

Integrating the last inequality from tot, we get kunðtÞk²_L2ðÞ4e^tkunðÞk²_L2ðÞþ

Zt

e^tskgð,sÞk²_L2ðÞdsþ2Mfðe^t1Þ 4e^tku_nðÞk²_L2ðÞþ2M_fjjðe^t1Þ

þ Zþ1

e^tskgð,sÞk²_L2ðÞdsþ Z þ2

þ1

e^tskgð,sÞk²_L2ðÞdsþ

4e^tkunðÞk²_L2ðÞþ2Mfjjðe^t1Þ þe^tð1þe¹þe²þ Þkgk²_L2 b

4e^tku_nðÞk²_L2ðÞþ2M_fjjðe^t1Þ þ e^t

1e¹kgk²_L2 b

:

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This allows us to state that {un} is bounded inL¹(,T;L²()), thanks to the fact that kun()kL²()4kukL²(). Integrating (3.1) on [,T], we have

kunðtÞk²_L2ðÞþ2 Z T

aðun,unÞdtþ2Mfkunk^q_LqðQ,TÞ

4ku_nðÞk_L2ðÞþ Z T

kgð,sÞk²_L2ðÞdsþ ku_nk²_L2ðQ,TÞþ2M_fjjðTÞ:

The last inequality implies that

fu_ngis bounded inL^qðQ_,TÞ, ð3:2Þ ZT

aðunðtÞ,unðtÞÞdt

is bounded: ð3:3Þ

Taking (H3) into account, we get the estimate Z

Q,T

jfðunÞj^q⁰4 Z

Q,T

Cð1þ junj^q1Þ^q⁰4 Z

Q,T

Cð1þ junj^qÞ:

Then {f(u_n)} is bounded inL^q⁰(Q_,T) and

fðu_nÞ* inL^q⁰ðQ_,TÞ: ð3:4Þ On the other hand, we rewrite the equation as

du_n

dt ¼aðu_n,Þ fðt,u_nÞ þgðx,tÞ and implement the following estimates:

ZT

aðun,vÞdt ¼

Z T

dt Z

’ðunÞdivððxÞrvÞ 4

ZT

dt Z

ðjunj^p1þ j’ð0ÞjÞjdivðrvÞj 4C

Z T

ku_nk

p p0

L^pðÞþ1 kvk_Vdt 4C

ku_nk

p p0

L^pðQ,TÞþ1

kvk_Lpð,T;VÞ, jhfðu_nÞ,vij4kfðu_nÞk_Lq0

ðQ,TÞkvk_LqðQ,TÞ, jhg,vij4kgk_Lq0

ðQ,TÞkvk_L^q_ðQ_,TÞ

for all v2L^p(,T;V)\L^q(Q,T). Then it follows that _du_n

dt is bounded in L^p⁰(,T;V⁰)þL^q⁰(Q,T). Combining this with (3.3) and using Proposition 2.4 we conclude that {u_n} is precompact inL^p(Q_,T). Hence we can assume that

. un!ustrongly in L^p(Q,T), . un!ua.e. in Q,T.

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Sincef2C(R), it follows thatf(un)!f(u) a.e. in Q,T. Thanks to (3.4), one has fðunÞ*fðuÞ weakly inL^q⁰ðQ,TÞ: ð3:5Þ Analogously, since {u_n} is bounded inL^p(Q,T), one can see that {’(u_n)} is bounded inL^p⁰(Q,T)L^p⁰(0,T;V⁰). Therefore’(un)*weakly inL^p⁰(0,T;V⁰). Putting this together with the fact that’(un)!’(u) a.e inQ,T, we have’(un)*’(u)¼weakly inL^p⁰(0,T;V⁰).

Finally, passing to the limit asn! 1in the relation ZT

hu_n⁰,vi þ Z T

dt Z

’ðunÞdivðrvÞ þ ZT

dt Z

fðunÞv¼ ZT

dt Z

gv

forv2L^p(,T;V)\L^q(Q,T), and using Proposition 2.5 we conclude thatuis a weak

solution of (1.1)–(1.3). g

3.2. Existence of the uniform attractor

Let us recall some definitions and related results. The pair of functions (f(s,), g(,s))¼(s) is called a symbol of (1.1). We consider (1.1) with a family of symbols including the shifted forms (sþh)¼(f(sþh,), g(,sþh)) and the limits of some sequence {(sþhn)}n2Nin an appropriate topological space. The family of such symbols is said to be the hull ofin and is denoted byH(), i.e.

HðÞ ¼clfðsþhÞh2Rg:

If the hullH() is a compact set in, we say that istranslation compactin . Let E be a Banach space (which in our case will be L²()), R_d¼{(t,)2 R²jt5} andP(E)¼{BEjB6¼ ;}. Assume that is a subspace of.

Definition 3.1 A family of mappings {U:R_dE! P(E)}2is called an MVP if there exists a continuous group {T(h) :!}h2R such that for all 2, x2E we have

(1) U(,,x)¼x, for all2R;

(2) U(t,,x)U(t,s,U(s,, x)) for allt5s5;

(3) U(tþh,þh,x)UT(h)(t,,x) for all (t,)2R_d,h2R. Denote by

C_q¼ f 2CðR;RÞ:j ðuÞj4C ð1þ juj^q1ÞforC 40g, k k_C_q ¼sup

u2R

j ðuÞj 1þ juj^q1:

ThenCqis a Banach space. We say thatfn!fin the spaceC(R;Cq) if

n!þ1lim sup

s2½t,tþr

kf_nðs,Þ fðs,Þk_C_q¼0 for allt2R,r40.

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Let f02CðR;CqÞ, g02L^2,w_locðR;L²ðÞÞand

Hðf0Þ ¼clCðR;CqÞff0ðsþhÞ jh2Rg, Hðg0Þ ¼cl_L^2,w

locðR;L²ðÞÞfg0ðsþhÞ jh2Rg,

where the topology in L^2,w_locðR;L²ðÞÞ is equipped by local weak convergence, i.e.

gn!g inL^2,w_locðR;L²ðÞÞif

n!þ1lim Ztþr

t

Z

ðg_nðx,sÞ gðx,sÞÞðx,sÞdxds¼0 for allt2R,r40 and2L²(Qt,tþr).

Let us take¼ H(f0) H(g0). For each¼(f,g)2, we define Uðt,,vÞ ¼ fu¼uðtÞ juis the solution of (1.1)(1.3), uðÞ ¼vg:

Then {U}2is an MVP with respect to the translation groupT(h)¼( þh) by the arguments in [22]. In addition,Uis a strict MVP, that isU(t,,x)¼U(t,s,U(s,,x)) for allt5s5.

Denote by

Uðt,,vÞ ¼[

2

Uðt,,vÞ andBR¼B(0,R), the ball inEcentred at 0 with radiusR.

Definition 3.2 The set AE is called a uniform attractor of MVP {U}2 if A6¼Eand

(1) Ais a uniformly attracting set, that is, for anyR40, for all2R, distðUðt,,B_RÞ,AÞ !0 ast! þ1;

(2) A is a minimal uniformly attracting set, that is, if is a uniformly attracting set thenAclE.

Here the notation dist(A,B):¼sup_a2Ainf_b2BkabkE, that is the Hausdorff semi-distance between two sets inE.

We use the following theorem as a sufficient condition for the existence of a uniform attractor described above. For a proof, see [18].

THEOREM 3.2 If the family of MVP{U}2satisfies the following conditions (1) there exists R040 such that, for all R40, dist(U(t, 0,BR),BR₀)!0 as

t! þ1,

(2) for all R40 and{tnjtn%þ1}, {njn2U(tn, 0,BR)}is precompact in E, then{U}2has a uniform attractor

A¼ \

s50

[

t5s

Uðt, 0,B_R₀_þ1Þ, which is compact in E.

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In order to deal with a uniform attractor, with respect to the family of symbols, one usually requires the translation compact property. Let us recall some discussions on this requirement. It is known that the hypothesis (H5) ensures thatgis translation compact inL^2,w_locðR;L²ðÞÞ, (see [8] for details). In addition, the following statement gives a sufficient condition for the translation compact property inC(R;Cq).

PROPOSITION3.3 [8] The function f2C(R; Cq)is translation compact if, and only if, for all R40 one has

(1) jf(t,v)j4C(R)for all t2R,v2[R,R],

(2) jf(t1,v1)f(t2,v2)j4(jt1t2j þ jv1v2j,R)for all t1,t22R;v1,v22[R,R].

Here C(R)40 and(s,R)!0as s!0^þ.

It is easy to see that, iff2C¹(R²;R) such thatjf(t,v)j4C(R), ^@f_@vðt,vÞ4CðRÞ and ^@f

@tðt,vÞ4CðRÞ for all t2R and for all v2[R,R] then the hypotheses in Proposition 3.3 are satisfied. For instance, the function f(t,u)¼ juj^q2u arctan t is well-checked.

From now on, we suppose that fis translation compact. Together with the fact thatg is translation compact inL^2,w_locðR;L²ðÞÞ, one has that is a compact set in CðR;C_qÞ L^2,w_locðR;L²ðÞÞ. Then it follows from [8] thatT(h) :!is continuous andT(h)for allh2R.

We need the following lemma to prove the dissipative property of MVP.

LEMMA 3.4 If u(t)is a weak solution of(1.1)–(1.3) then

kuðtÞk²_L2ðÞ4e^ðtÞkuk²_L2ðÞþM0ð1e^ðtÞÞ þ kgk²_L2

b

1e¹, where M0¼M0(q,jj,Mf)40.

Proof From (1.1), using (H4) and Cauchy inequality, we have d

dtkuðtÞk²_L2ðÞþ2aðu,uÞ þ2Mfkuk^q_LqðÞ

42Mfjj þ kuðtÞk²_L2ðÞþ kgðtÞk²_L2ðÞ: ð3:6Þ Sinceq42, using Young’s inequality, one has

2kuðtÞk²_L2ðÞ4CkuðtÞk²_LqðÞ42M_fkuðtÞk^q_LqðÞþM_q, whereMq40. Putting this into (3.6), we obtain

d

dtkuðtÞk²_L2ðÞþ kuðtÞk²_L2ðÞ42Mfjj þMqþ kgðtÞk²_L2ðÞ: Then

d

dtðe^tkuðtÞk²_L2ðÞÞ4e^tð2Mfjj þMqÞ þe^tkgðtÞk²_L2ðÞ:

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Taking integration over [,t], we arrive at

kuðtÞk²_L2ðÞ4e^ðtÞku_nk²_L2ðÞþ ð1e^ðtÞÞð2M_fjj þM_qÞ þ

Zt

e^ðtsÞkgðsÞk²_L2ðÞds

4e^ðtÞkunk²_L2ðÞþ ð1e^ðtÞÞð2Mfjj þMqÞ þ

Zt t1

e^ðtsÞkgðsÞk²_L2ðÞdsþ Z t1

t2

e^ðtsÞkgðsÞk²_L2ðÞdsþ 4e^ðtÞkunk²_L2ðÞþ ð1e^ðtÞÞð2Mfjj þMqÞ

þ ð1þe¹þe²þ Þsup

t2R

kgðsÞk²_L2ðÞ

4e^ðtÞkunk²_L2ðÞþ ð1e^ðtÞÞð2Mfjj þMqÞ þ kgk²_L2

b

1e¹:

So we complete the proof. g

LEMMA 3.5 Let the hypotheses(H1)–(H5)hold.Assume that{un}n2Nis a sequence of weak solutions of (1.1)–(1.3) with respect to the sequence of symbols {n}_n2,n2N

such that

(1) un()*u in L²(), (2) n! in,

then there exists a solution u of (1.1)–(1.3) with respect to the symbol such that u()¼u and un(t)!u(t)in L²()for any t4.

Proof We will adapt the technique as in [13,22] to prove this statement. Let n¼(fn,gn). Sincefsatisfies (H3)–(H4) for allt2R andfn2 H(f), one sees thatfn, n2N also satisfies (H3)–(H4) with the same constants Cf and Mf. From (1.1), we have

d

dtku_nðtÞk²_L2ðÞþaðu_n,u_nÞ þ2M_fku_nk^q_LqðÞ

42Mfjj þ kgnðtÞk²_L2ðÞþ kunðtÞk²_L2ðÞ

for allt4. Using the same arguments as in the proof of Lemma 3.4, we have ku_nðtÞk²_L2ðÞ4e^ðtÞku_nðÞk²_L2ðÞþM₀ð1e^ðtÞÞ þ 1

1e¹kg_nk²_L2 b

: Since {un()} is bounded and kg_nk²_L2

b

4kgk²_L2 b

, we have {un} is bounded in L¹(0,T;L²()) for givenT4. In particular,

u_nðtÞ*uðtÞ inL²ðÞfor eacht2 ½,T ð3:7Þ up to a subsequence. Now using the arguments as in the proof of Theorem 3.1, we have

. RT

aðun,unÞdtis bounded, . {un} is bounded inL^q(Q,T),

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. {fn(un)} is bounded inL^q⁰(Q,T),

. fu_n⁰g is bounded inL^p⁰(,T;V⁰)þL^q⁰(Q,T).

Using Proposition 2.4, one gets that {un} is precompact in L^p(Q,T) and then, by replacing {un} with a subsequence if necessary, we have

. un!ua.e in Q,T,

. ’(un)*’(u) inL^p⁰(,T;V⁰), . u_n⁰*u⁰in L^p⁰(,T;V⁰)þL^q⁰(Q,T).

Let n !¼ ðf,gÞ in. In order to prove that u is a weak solution of (1.1)–(1.3) with respect to the symbol, we need to pass to the limit in following relation:

ZT

hu_n⁰,vi þ ZT

dt Z

’ðu_nÞdivðrvÞ þ Z T

dt Z

f_nðu_nÞv¼ Z T

dt Z

g_nv forv2L^p(,T;V)\L^q(Q,T). Sincegn*ginL²(,T;L²()), it remains to prove that f_nðu_nÞ*fðuÞ inL^q⁰(Q,T). We have the stronger claim, that isf_nðu_nÞ !fðuÞ strongly inL^q⁰(Q,T). Indeed,

Z T

Z

jf_nðt,u_nÞ fðt, uÞj^q⁰dxdt42^q⁰ ZT

Z

jfðt,u_nÞ fðt,uÞj^q⁰dxdt þ2^q⁰

ZT

Z

jfnðt,unÞ fðt, unÞj^q⁰

ð1þ ju_nj^q1Þ^q⁰ ð1þ junj^q1Þ^q⁰dxdt 42^q⁰

ZT

Z

jfðt,unÞ fðt,uÞj^q⁰dxdt þ

4 sup

½,T

jf_nfj_C_q q⁰Z T

Z

ð1þ ju_nj^qÞdxdt:

Then the boundedness of {un} in L^q(Q,T) and the continuity of fguarantee our claim. In addition, by Proposition 2.5, we getu2C([,T]; L²()). Then the initial condition makes sense.

We are in a position to show thatun(t)!u(t) inL²() for anyt4. Taking into account of (3.7), we have to check thatkun(t)kL²()! ku(t)kL²()inR.

Let us denote

JnðtÞ ¼ kunðtÞk²_L2ðÞ2 Zt

ðgnðsÞ,unðsÞÞ_L2ðÞdtMjjðtÞ, JðtÞ ¼ kuðtÞk²_L2ðÞ2

Zt

ðgðsÞ,uðsÞÞ_L2ðÞdtMjjðtÞ

for someM40. ThenJn,J2C([,T];R). Arguing as in the proof of Lemma 3.4,un

satisfies the estimate d

dtkunðtÞk²_L2ðÞ4Mjj þ2ðgnðtÞ,unðtÞÞ_L2ðÞ,

and the same estimate is valid foru. Hence,JnandJare decreasing on [,T]. We first show that

JnðtÞ !JðtÞfor a.e.t2 ½,T: ð3:8Þ

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Obviously

JnðtÞ JðtÞ

4ku_nðtÞk²_L2ðÞ kuðtÞk²_L2ðÞbigj þ2 Zt

ðgn,unÞ_L2ðÞ ðg,uÞ_L2ðÞdt 4kunðtÞ uðtÞk_L²_ðÞðkunðtÞk_L²_ðÞþ kuðtÞk_L²_ðÞÞ

þ2 Zt

ðgn,unuÞ_L²_ðÞdt þ2

Z t

ðgng,uÞ_L²_ðÞdt : Moreover, one has

Zt

ðg_n,u_nuÞ_L2ðÞdt4kg_nk_L2ðQ,tÞku_nuk_L2ðQ,tÞ!0

as n! 1 since un!u strongly in L²(Q,t) and {gn} is bounded in L²(Q,t). In addition

Zt

ðgng,uÞ_L2ðÞdt!0

as n! 1 since gn*g in L²(Q,t). Then (3.8) is proved due to the fact that un(t)!u(t) inL²() for a.e.t2(,T).

Suppose that {tm} is an increasing sequence in [,T] such thattm!tasm! 1.

Then

. Jn(tm)!Jn(t) asm! 1, . Jn(tm)!J(tm) asn! 1.

So for40, we have eventually

JnðtÞ JðtÞ4JnðtmÞ JðtÞ ¼JnðtmÞ JðtmÞ þJðtmÞ JðtÞ5":

Similarly,J(t)Jn(t)5". ThereforeJn(t)!J(t) and then

kunðtÞk_L2ðÞ! kuðtÞk_L2ðÞ asn! 1: g The following theorem is the main result in this section.

THEOREM 3.6 Under the hypotheses(H1)–(H5),the MVP{U}2generated by the problem(1.1)–(1.3) possesses a uniform attractor which is a compact set in L²().

Proof Note that each symbol n¼(fn, gn)2 satisfies the same conditions as in (H3)–(H5). Furthermore, sincegn2 H(g), we havekgnk_L2

b4kgk_L²

b. Hence it follows from Lemma 3.4 that, if unis the weak solution of (1.1)–(1.3) with respect to the symboln, one has

kunðtÞk²_L2ðÞ4e^ðtÞkunðÞk²_L2ðÞþM0ð1e^ðtÞÞ þ kgk²_L2

b

1e¹:

The last inequality ensures that, ifun()2BRthen there existsT0¼T0(,R) such that unðtÞ 2BR0, for allt5T0,

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where

R0¼2M0þ kgk²_L2

b

1e¹:

That is U(t, , BR)BR₀ for all t5T0(,R). This fulfils the first condition in Theorem 3.2. We now verify the second condition.

Lettn%þ1andn2U(tn, 0,BR). Then there exists a sequence of solutions {un} of (1.1)–(1.3) with respect to the sequence of symbols {n} such thatn¼un(tn). For givent40, we have

U_nðt_n, 0,B_RÞ ¼U_nðtþt_nt, 0,B_RÞ

Unðtþtnt,tnt,Unðtnt, 0,BRÞÞ U_nðtþt_nt,t_nt,B_R₀Þ

fortn5T0þt. Hence

U_nðt_n, 0,B_RÞ U_Tðt_n_t_Þ_nðt, 0,B_R₀Þ:

Now_n2U_nðt, 0,B_R₀Þwhere_n ¼Tðt_ntÞ_n2. That is,n¼vn(t) wherevn

is the weak solution of (1.1)–(1.3) with respect to the symbol _n. Suppose that _n!in. By Lemma 3.5, there exists a solutionvof (1.1)–(1.3) with respect to the symbol, such thatn¼vn(t)!v(t) inL²() and thus the proof completes. g

Acknowledgements

This work is supported in part by Taiwan NSC grant (NSC96-2115-M-110-004-MY3).

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