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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 312536, 17 pages
http://dx.doi.org/10.1155/2012/312536
Research Article

Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

1School of Mathematical Sciences, Anhui University, Hefei 230039, China
2School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou 730070, China

Received 10 July 2012; Accepted 17 August 2012

Academic Editor: Sergey V. Zelik

Copyright © 2012 Weisheng Niu and Hongtao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Ω be a smooth bounded domain in 𝑁,(𝑁3). We consider the asymptotic behavior of solutions to the following problem 𝑢𝑡div(𝑎(𝑥)𝑢)+𝜆𝑓(𝑢)=𝜇inΩ×+,𝑢=0on𝜕Ω×+,𝑢(𝑥,0)=𝑢0(𝑥)inΩ, where 𝑢0𝐿1(Ω), 𝜇 is a finite Radon measure independent of time. We provide the existence and uniqueness results on the approximated solutions. Then we establish some regularity results on the solutions and consider the long-time behavior.

1. Introduction

We consider the asymptotic behavior of solutions to the following equations 𝑢𝑡div(𝑎(𝑥)𝑢)+𝜆𝑓(𝑢)=𝜇,inΩ×+,𝑢=0,on𝜕Ω×+,𝑢(𝑥,0)=𝑢0(𝑥),inΩ,(1.1) where Ω is a bounded domain in 𝑁(𝑁2) with smooth boundary 𝜕Ω, 𝑢0𝐿1(Ω), 𝜆0,𝜇 is a finite Radon measure independent of time, 𝑎(𝑥) is a matrix with bounded, measurable entries, and satisfying the ellipticity assumption ||𝜉||𝑎(𝑥)𝜉𝜉𝑐2,forany𝜉N,with𝑐>0.(1.2) Concerning the nonlinear term, we assume that 𝑓 is a 𝐶1 function satisfying, 𝑓||𝑓(𝑠)𝑙,(1.3)||(𝑠)𝐶1+|𝑠|𝑝2,𝑝2,(1.4)𝐶|𝑠|𝑝𝑘𝑓(𝑠)𝑠𝐶|𝑠|𝑝+𝑘,𝑝2,(1.5) both for all 𝑠𝑅, where 𝑙,𝐶,𝑘 are positive constants.

Parabolic equations with 𝐿1 or measure data arise in many physical models, control problems, and in models of turbulent flows in oceanography and climatology [14]. Existence and regularity results for parabolic equations with 𝐿1 and measure data have been studied widely by many authors in the past decades, see [510]. The usual approach to study problems with these kinds of data is approximation. The basic reference for these arguments is [7], where the authors obtained weak solutions (in distribution sense) to nonlinear parabolic equations. In our setting, such a solution is a function 𝑢𝐿1((0,𝑇);𝑊01,1(Ω)) such that 𝑓(𝑢)𝐿1(𝑄𝑇) for any 𝑇>0, and 𝑄𝑇𝑢𝜑𝑡𝑑𝑥𝑑𝑡+𝑄𝑇𝑎(𝑥)𝑢𝜑𝑑𝑥𝑑𝑡+𝜆𝑄𝑇𝑓(𝑢)𝜑𝑑𝑥𝑑𝑡=𝑄𝑇𝜑𝑑𝜇+Ω𝑢0(𝑥)𝜑(0,𝑥)𝑑𝑥,(1.6) for any 𝜑𝐶𝑐([0,𝑇)×Ω).

Generally, the regularity of weak solutions (in distribution sense) is not strong enough to ensure uniqueness [8]. But one may select a weak solution which is “better” than the others. Since one may prove that the weak solution obtained from approximation does not depend on the approximation chosen for the irregular data. In such a sense, it is the only weak solution which is found by means of approximations; we may call it approximated solutions. Such a concept was first introduced by [9]. Here in the present paper, we will focus ourselves to the scope of approximated solutions, that is, weak solutions obtained as limits of approximations.

The long-time behavior of parabolic problems with irregular data (such as 𝐿1 data, measure data) have been considered by many authors [1116]. In [11, 12], existence of global attractors for porous media equations and m-Laplacian equations with irregular initial data were deeply studied, while in [13, 14] the convergence to the equilibrium for the solutions of parabolic problems with measrued data were thoroughly investigated. In [15, 16], we considered the existence of global attractors for the parabolic equations with 𝐿1 data.

In this paper, we intend to consider the asymptotic behavior of approximated solutions to problem (1.1) with measure data. Precisely speaking, we assume that the forcing term in the equations is just a finite Radon measure. For the case 𝜆>0, to ensure the existence result for large 𝑝 in (1.5) [17], we restrict ourselves to diffuse measures, that is, 𝜇 does not charge the sets of zero parabolic 2-capacity (see details for parabolic 𝑝-capacity in [18]). We first provide the existence result for problem (1.1) and prove the uniqueness of the approximated solution. Then using some decomposition techniques, we establish some new regularity results and show the existence of a global attractor 𝒜 in 𝐿𝑝1(Ω)𝑊01,𝑞(Ω) with 𝑞<max{𝑁/(𝑁1),(2𝑝2)/𝑝}, which attracts every bounded subset of 𝐿1(Ω) in the norm of 𝐿𝑟(Ω)𝐻10(Ω), for any 𝑟[1,).

For the case 𝜆=0, we consider general bounded Radon measure 𝜇 which is independent of time. We provide the uniqueness of approximated solutions for the parabolic problem and its corresponding elliptic problem. Then we prove that the approximated solution of the parabolic equations converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of 𝐿𝑟(Ω)𝐻10(Ω), for any 𝑟[1,), though they all lie in some less regular spaces.

Our main results can be stated as follows.

Theorem 1.1. Assume that 𝑢0𝐿1(Ω), 𝜆>0, 𝜇 is a bounded Radon measure, which does not charge the sets of zero parabolic 2-capacity and is independent of time, 𝑓 is a 𝐶1 function satisfying assumptions (1.3)–(1.5). Then the semigroup {𝑆(𝑡)}𝑡0, generated by approximated solutions of problem (1.1), possesses a global attractor 𝒜 in 𝐿1(Ω). Moreover, 𝒜 is compact and invariant in 𝐿𝑝1(Ω)𝑊01,𝑞(Ω) with 𝑞<max{𝑁/(𝑁1),(2𝑝2)/𝑝}, and attracts every bounded subset of 𝐿1(Ω) in the norm topology of 𝐿𝑟(Ω)𝐻10(Ω), 1𝑟<.

Theorem 1.2. Assume that 𝑢0𝐿1(Ω), 𝜆=0, 𝜇 is a bounded Radon measure independent of time. Then the approximated solution 𝑢(𝑡) of problem (1.1) is unique and converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of 𝐿𝑟(Ω)𝐻10(Ω), for any 1𝑟<.

Remark 1.3. Though 𝑢(𝑡) and 𝑣 all lie in some less-regular spaces, 𝑢(𝑡) converges to 𝑣 in stronger norm, that is, 𝑢(𝑡)𝑣 converges to 0 in 𝐿𝑟(Ω)𝐻10(Ω), 1𝑟<. Such a result, in some sense, sharpens the result of [13], where the author showed that 𝑢(𝑡) converges to 𝑣 in 𝐿1(Ω).

We organize the paper as follows: in Section 2, we provide the existence of approximated solutions, prove the uniqueness result and some useful lemmas; in Section 3, we establish some improved regularity results on the approximated solutions. At last, in Section 4, we prove the main theorems.

For convenience, for any 𝑇>0 we use 𝑄𝑇 to denote Ω×(0,𝑇) hereafter. Also, we denote by |𝐸| the Lebesgue measure of the set 𝐸, and denote by 𝐶 any positive constant which may be different from each other even in the same line.

2. Existence Results and Useful Lemmas

We begin this section by providing some existence results on the approximated solutions.

Definition 2.1. A function 𝑢 is called an approximated solution of problem (1.1), if 𝑢𝐿1((0,𝑇);𝑊01,1(Ω)), 𝑓(𝑢)𝐿1(𝑄𝑇) for any 𝑇>0, and 𝑄𝑇𝑢𝜑𝑡𝑑𝑥𝑑𝑡+𝑄𝑇𝑎(𝑥)𝑢𝜑𝑑𝑥𝑑𝑡+𝜆𝑄𝑇𝑓(𝑢)𝜑𝑑𝑥𝑑𝑡=𝑄𝑇𝜑𝑑𝜇+Ω𝑢0(𝑥)𝜑(0,𝑥)𝑑𝑥,(2.1) for any 𝜑𝐶𝑐([0,𝑇)×Ω), and moreover, 𝑢 is obtained as limit of solutions to the following approximated problem 𝑢𝑛𝑡div(𝑎(𝑥)𝑢𝑛)+𝜆𝑓(𝑢𝑛)=𝜇𝑛,inΩ×+,𝑢=0,on𝜕Ω×+,𝑢(𝑥,0)=𝑢𝑛0,inΩ,(2.2) where {𝜇𝑛},{𝑢𝑛0} is a smooth approximation of data 𝜇,𝑢0.

Theorem 2.2. Under the assumptions of Theorem 1.1, problem (1.1) has a unique approximated solution 𝑢𝐶([0,𝑇];𝐿1(Ω)), for any 𝑇>0, satisfying (i) if 𝑝(2𝑁+2)/𝑁, then 𝑢𝐿𝑞((0,𝑇);𝑊01,𝑞(Ω)) with 𝑞<(2𝑝2)/𝑝;(ii) if 2𝑝<(2𝑁+2)/𝑁, then 𝑢𝐿𝑞((0,𝑇);𝑊01,𝑞(Ω)) with 𝑞<(𝑁+2)/(𝑁+1).

Proof. According to [18, Theorem 2.12], if a Radon measure 𝜇 on 𝑄𝑇 does not charge the sets of zero parabolic 2-capacity and is independent of time, 𝜇 can actually be identified as a Radon measure which is absolutely continuous with respect to the elliptic 2-capacity. Using Theorem 2.1 of [19], 𝜇 can be decomposed as 𝜇=𝑔+div𝐺, where 𝑔𝐿1(Ω),𝐺(𝐿2(Ω))𝑁. Hence, we need only to consider the following problem 𝑢𝑡div(𝑎(𝑥)𝑢)+𝜆𝑓(𝑢)=𝑔+div𝐺,inΩ×+,𝑢=0,on𝜕Ω×+,𝑢(𝑥,0)=𝑢0,inΩ.(2.3) The proof of existence part of the theorem is similar to [9]. Besides, one can prove 𝑢𝐶([0,𝑇];𝐿1(Ω)) using arguments similar to CLAIM 2 in [8]. So we omit the details of them and only prove the uniqueness result.
Let {𝑔𝑛}𝑛, {𝑢𝑛0}𝑛 be a smooth approximation of data 𝑔 and 𝑢0 with 𝑢𝑛0𝐿1(Ω)𝑢0𝐿1(Ω),̃𝑔𝑛𝐿1(Ω)𝑔𝐿1(Ω),(2.4) and let {̃𝑔𝑛}𝑛, {̃𝑢𝑛0}𝑛 be another smooth approximation of the data with ̃𝑢𝑛0𝐿1(Ω)𝑢0𝐿1(Ω),̃𝑔𝑛𝐿1(Ω)𝑔𝐿1(Ω).(2.5) Assume that 𝑢,̃𝑢 are two approximated solutions to problem (1.1), obtained as limit of the solutions to the following two approximated problems, respectively, 𝑢𝑛𝑡div(𝑎(𝑥)𝑢𝑛)+𝜆𝑓(𝑢𝑛)=𝑔𝑛+div𝐺,inΩ×+,𝑢=0,on𝜕Ω×+,𝑢(𝑥,0)=𝑢𝑛0,inΩ,(2.6)̃𝑢𝑛𝑡div(𝑎(𝑥)̃𝑢𝑛)+𝜆𝑓(̃𝑢𝑛)=̃𝑔𝑛+div𝐺,inΩ×+,𝑢=0,on𝜕Ω×+,𝑢(𝑥,0)=̃𝑢𝑛0,inΩ.(2.7) Now we prove that 𝑢=̃𝑢. For any 𝑘>0, define 𝜓𝑘(𝑠) as 𝜓𝑘(𝑠)=𝑘,𝑠>𝑘𝑠,|𝑠|𝑘𝑘,𝑠<𝑘.(2.8) Let Ψ𝑘(𝜎)=𝜎0𝜓𝑘(𝑠)𝑑𝑠 be its primitive function. Taking 𝜓𝑘(𝑢𝑛̃𝑢𝑛) as a test function in (2.6) and (2.7), we deduce that ΩΨ𝑘(𝑢𝑛̃𝑢𝑛)(𝑡)𝑑𝑥ΩΨ𝑘𝑢𝑛0̃𝑢𝑛0𝑑𝑥+𝑄𝑇𝑎(𝑥)𝜓𝑘(𝑢𝑛̃𝑢𝑛)2𝑑𝑥𝑑𝑡=𝜆𝑄𝑇(𝑓(̃𝑢𝑛)𝑓(𝑢𝑛))𝜓𝑘(𝑢𝑛̃𝑢𝑛)𝑑𝑥𝑑𝑡+𝑄𝑇(𝑔𝑛̃𝑔𝑛)𝜓𝑘(𝑢𝑛̃𝑢𝑛)𝑑𝑥𝑑𝑡𝑄𝑇𝜆𝑙(𝑢𝑛̃𝑢𝑛)𝜓𝑘(𝑢𝑛̃𝑢𝑛)𝑑𝑥𝑑𝑡+𝑄𝑇||𝑔𝑛̃𝑔𝑛||𝑑𝑥𝑑𝑡.(2.9) Hence, from the assumptions on 𝑓, we get ΩΨ𝑘(𝑢𝑛̃𝑢𝑛)(𝑡)𝑑𝑥2𝑙𝜆𝑇0ΩΨ𝑘(𝑢𝑛̃𝑢𝑛)𝑑𝑥𝑑𝑡+𝑄𝑇||𝑔𝑛̃𝑔𝑛||𝑑𝑥𝑑𝑡+ΩΨ𝑘𝑢𝑛0̃𝑢𝑛0𝑑𝑥.(2.10) Let 𝑛, we have ΩΨ𝑘(𝑢̃𝑢)(𝑡)𝑑𝑥2𝑙𝜆𝑇0ΩΨ𝑘(𝑢̃𝑢)(𝑡)𝑑𝑥𝑑𝑡.(2.11) Thus for all 𝑘>0, we have sup[]0,𝑇ΩΨ𝑘(̃𝑢𝑢)(𝑡)𝑑𝑥2𝑙𝑇sup[]0,𝑇ΩΨ𝑘(̃𝑢𝑢)𝑑𝑥.(2.12) Taking 𝑇 small enough such that 2𝑙𝜆𝑇<1, we deduce that Ψ𝑘(𝑢𝑢)=0 for all 𝑘>0 in 𝑄𝑇, thus 𝑢𝑢 in 𝑄𝑇. Dividing [0,𝑇] into several intervals to carry out the same arguments, we obtain the uniqueness of the approximated solution.
Similar to [20], we can prove the following.

Theorem 2.3. Under the assumptions of Theorem 1.1, there exists at least one approximated solution 𝑣 to the stationary problem of corresponding to problem (1.1), with (i)if 𝑝(2𝑁2)/(𝑁2), we have 𝑣𝑊01,𝑞(Ω) for 𝑞<(2𝑝2)/𝑝;(ii)if 2𝑝<(2𝑁2)/(𝑁2), we have 𝑣𝑊01,𝑞(Ω) for 𝑞<𝑁/(𝑁1).

Remark 2.4. Note that if 𝑣 is an approximated solution to following problem div(𝑎(𝑥)𝑣)+𝜆𝑓(𝑣)=𝑔+div𝐺,inΩ,𝑣=0,on𝜕Ω,(2.13) then there is a sequence {𝑣𝑛} converges to 𝑣, where 𝑣𝑛 is the solution of the corresponding approximated problem div(𝑎(𝑥)𝑣𝑛)+𝜆𝑓(𝑣𝑛)=𝑔𝑛𝑣+div𝐺,inΩ,𝑛=0,on𝜕Ω.(2.14) And hence 𝑣𝑛 is a solution of parabolic equations 𝑣𝑛𝑡div(𝑎(𝑥)𝑣𝑛)+𝜆𝑓(𝑣𝑛)=𝑔𝑛+div𝐺,inΩ×+,𝑣𝑛=0,on𝜕Ω×+,𝑣𝑛(0)=𝑣𝑛(𝑥),inΩ.(2.15) Thus, 𝑣 is an approximated solution of problem (1.1) with initial data 𝑢0=𝑣(𝑥).

Under the assumptions of Theorem 1.2, the problem turns out to be 𝑢𝑡div(𝑎(𝑥)𝑢)=𝜇,inΩ×+,𝑢=0,on𝜕Ω×+,𝑢(𝑥,0)=𝑢0,inΩ.(2.16) The existence of approximated solutions to problem (1.1) and the elliptic equations corresponding to it follows directly from Sections IV and II of [7]. Form Section 2.3 of [21], we know that the approximated solution to the stationary equations is actually a duality solution, and hence unique. Furthermore, it is not difficult to prove that an approximated solution for linear parabolic equations turns out to be a duality solution, and hence unique too.

Lemma 2.5. Under the assumptions of Theorem 1.2, an approximated solution to the parabolic problem (1.1), (2.16) turns out to be a duality solution, and conversely.

Proof. The proof is mainly similar to that of Theorem 6 in [22]. We just sketch it. Let 𝑢 be an approximated solution, then there exist a smooth approximation {𝜇𝑛}𝑛, {𝑢𝑛0}𝑛 of data 𝜇 and 𝑢0, such that the solution of the approximated problem of (2.16) with data 𝜇𝑛 and 𝑢𝑛0 converges to 𝑢. Let 𝐶𝑐(𝑄𝑇) and 𝜔 be the solution of the the following parabolic problem 𝜔𝑡𝑎div(𝑥)𝜔=,inΩ×(0,𝑇),𝜔=0,on𝜕Ω×(0,𝑇),𝜔(𝑥,𝑇)=0,inΩ,(2.17) where 𝑎(𝑥) is the transposed matrix of 𝑎(𝑥). Taking 𝜔 as a test function in the approximated problem and taking 𝑢𝑛 as a test function in the problem above, then let 𝑛 go to infinity we obtain that the approximated solution is a duality solution. Form the uniqueness of duality solutions [13], we get the conclusion.

Now we provide two lemmas which are useful in analyzing the regularity and asymptotic behavior of the solutions to problem (1.1).

Lemma 2.6 (see [15]). Let 𝑋, 𝑌 be two Banach spaces, let 𝑋 be separable, reflexive, and let 𝑋𝑌 with dual 𝑋. Suppose that {𝑢𝑛} is uniformly bounded in 𝐿((0,𝑇);𝑋) with esssup𝑡[0,𝑇]𝑢𝑛(𝑡)𝑋𝐶,(2.18) and that 𝑢𝑛𝑢 weakly in 𝐿𝑟((0,𝑇);𝑋) for some 𝑟(1,). Then esssup𝑡[0,𝑇]𝑢(𝑡)𝑋𝐶.(2.19) Moreover, if 𝑢𝐶([0,𝑇];𝑌), then in fact sup𝑡[0,𝑇]𝑢(𝑡)𝑋𝐶.(2.20)

Lemma 2.7 (see [16]). Let 𝑋,𝑌 be two Banach spaces with imbedding 𝑋𝑌, let {𝑆(𝑡)}𝑡0 be a continuous semigroup on 𝑌. Assume that {𝑆(𝑡)}𝑡0 is asymptotically compact in 𝑋 and has an absorbing set 𝐵0𝑋, that is, for any bounded set 𝐵𝑌, there exists a 𝑇=𝑇(𝐵) such that 𝑆(𝑡)𝐵𝐵0,𝑡𝑇.(2.21) Then {𝑆(𝑡)}𝑡0 has a global attractor 𝒜 in 𝑋, which is compact, invariant in 𝑋 and attracts every bounded sets of 𝑌 in the topology of 𝑋.

Remark 2.8. To obtain global attractors, one usually needs the semigroup to be norm-to-norm continuous, weak-to-weak, or norm-to-weak continuous [2327]. Here, to obtain global attractors in the space 𝑋, we need neither of them. We only need the semigroup to be continuous in a less-regular space 𝑌.

3. Improved Regularity Results on the Approximated Solutions

In this section, we prove the following regularity results on the approximated solution 𝑢 to problem (1.1).

Theorem 3.1. Under the assumptions of Theorem 2.2, let 𝑢(𝑡) be the approximated solution to problem (1.1). Then 𝑢 admits the decomposition 𝑢(𝑥,𝑡)=𝑤(𝑥,𝑡)+𝑣(𝑥), with 𝑣 being an approximated solution to problem (2.13), and 𝑤 being an approximated solution of the following problem 𝑤𝑡div(𝑎(𝑥)𝑤)+𝜆𝑓(𝑣+𝑤)𝜆𝑓(𝑣)=0,inΩ×+,𝑤=0,on𝜕Ω×+,𝑤(𝑥,0)=𝑢0𝑣,inΩ.(3.1) Moreover, we have (i)𝑤𝐿((𝛿,𝑇);𝐿𝑞(Ω)) for any 0<𝛿<𝑇,1𝑞<. Moreover, there exists a constant 𝑀𝑞 and a time 𝑡𝑞(𝑢0,𝑔,𝐺) such that 𝑤(𝑡)𝐿𝑞(Ω)𝑀𝑞forall𝑡𝑡𝑞(𝑢0,𝑔,𝐺).(ii)𝑤𝐿((𝛿,𝑇);𝐻10(Ω)) for any 0<𝛿<𝑇. Moreover, there exists a constant 𝜌 and a time 𝑇0(𝑢0,𝑔,𝐺) such that 𝑤(𝑡)𝐻10(Ω)𝜌forall𝑡𝑇0(𝑢0,𝑔,𝐺).

Proof. We follow the lines of [15, 28]. Let {𝑔𝑛} be a sequence of smooth data which converges to 𝑔 in 𝐿1(Ω) and 𝑔𝑛𝐿1(Ω)𝑔𝐿1(Ω). Let 𝑣𝑛 be a solution of the following approximated problem for each 𝑛, div(𝑎(𝑥)𝑣𝑛)+𝜆𝑓(𝑣𝑛)=𝑔𝑛𝑣+div𝐺,inΩ,𝑛=0,on𝜕Ω.(3.2) Then 𝑣𝑛 converges (up to subsequences) to an approximated solution 𝑣 strongly in 𝐿1(Ω), and weakly in 𝑊01,𝑞(Ω),1𝑞<𝑁/(𝑁1). Let {𝑢𝑛} be a sequence of solutions to the following approximated problem 𝑢𝑛𝑡div(𝑎(𝑥)𝑢𝑛)+𝜆𝑓(𝑢𝑛)=𝑔𝑛(𝑥)+div𝐺,inΩ×+,𝑢𝑛(𝑥)=0,on𝜕Ω×+,𝑢𝑛(𝑥,0)=𝑢𝑛0,inΩ,(3.3) where 𝑢𝑛0 converges to 𝑢0 with 𝑢𝑛0𝐿1(Ω)𝑢0𝐿1(Ω). Similar to [8, 29], we know that 𝑢𝑛𝑢weaklyin𝐿𝑞(0,𝑇);𝑊01,𝑞(Ω),𝑞<𝑁+2,𝑢𝑁+1𝑛[]𝑢in𝐶0,𝑇;𝐿1.(Ω)(3.4) Now let 𝑤𝑛(𝑡)=𝑢𝑛(𝑡)𝑣𝑛. Then 𝑤𝑛 satisfies 𝑤𝑛𝑡div(𝑎(𝑥)𝑤𝑛)+𝜆𝑓(𝑣𝑛+𝑤𝑛)𝜆𝑓(𝑣𝑛)=0,inΩ×+,𝑤𝑛=0,on𝜕Ω×+,𝑤𝑛(𝑥,0)=𝑢𝑛0𝑣𝑛,inΩ.(3.5) Similarly, we have 𝑤𝑛 (up to subsequences) converges to the approximated solution 𝑤 of problem (3.1) in 𝐶([0,𝑇];𝐿1(Ω)) and weakly in 𝐿𝑞((0,𝑇);𝑊01,𝑞(Ω)),𝑞<(𝑁+2)/(𝑁+1).
Now we prove (i). Taking 𝜓1(𝑢𝑛) as test function in (3.3) (for simplicity we take 𝜆=1), we deduce that 𝑑𝑑𝑡ΩΨ1(𝑢𝑛)𝑑𝑥+Ω𝑓(𝑢𝑛)𝜓1(𝑢𝑛)𝑑𝑥𝑔𝐿1(Ω)+𝐶𝐺𝐿2(Ω).(3.6) Since 𝑓(𝑢𝑛)𝜓1(𝑢𝑛𝐶||𝑢)𝑛||𝑝1||𝜓𝐶1(𝑢𝑛)||,(3.7) we have 𝑑𝑑𝑡ΩΨ1(𝑢𝑛)𝑑𝑥+𝐶Ω||Ψ1(𝑢𝑛)||𝑑𝑥𝑔𝐿1(Ω)+𝐶𝐺𝐿2(Ω)||Ω||.+𝐶(3.8) The Gronwall's inequality implies that ΩΨ1(𝑢𝑛(𝑢𝑡))𝑑𝑥0𝐿1(Ω)𝑒𝐶𝑡||Ω||+𝐶+𝐶𝑔𝐿1(Ω)+𝐶𝐺𝐿2(Ω).(3.9) Noticing that Ω||𝑢𝑛(||𝑡)𝑑𝑥ΩΨ1(𝑢𝑛(||Ω||,𝑡))𝑑𝑥+(3.10) we obtain that Ω||𝑢𝑛(||𝑢𝑡)𝑑𝑥0𝐿1(Ω)𝑒𝐶𝑡||Ω||+𝐶+𝐶𝑔𝐿1(Ω)+𝐶𝐺𝐿2(Ω),𝑡0.(3.11) Moreover, integrating (3.6) between 𝑡 and 𝑡+1 and using (3.7) we have 𝑡𝑡+1Ω||𝑢𝑛||𝑝1𝑢𝑑𝑥𝑑𝜉𝐶0𝐿1(Ω)𝑒𝐶𝑡+||Ω||+𝑔𝐿1(Ω)+𝐺𝐿2(Ω).(3.12) Similarly, taking 𝜓1(𝑣𝑛) as test function in (3.2), we can deduce that Ω||𝑓(𝑣𝑛)||𝑑𝑥𝐶Ω||𝑣𝑛||𝑝1+1𝑑𝑥𝐶𝑔𝐿1(Ω)+𝐺𝐿2(Ω)+||Ω||.(3.13) Hence, 𝑡𝑡+1Ω||𝑤𝑛||𝑝1𝑑𝑥𝑑𝜉𝐶𝑔𝐿1(Ω)+𝐺𝐿2(Ω)+𝑢0𝐿1(Ω)+||Ω||,(3.14) with 𝐶 independent of 𝑛, for 𝑡0.
Now we use bootstrap method in the case 𝑝3. The case 2𝑝<3 can be treated similarly with minor modifications. Multiplying (3.5) by |𝑤𝑛|𝑞02𝑤𝑛,𝑞0=𝑝12, and integrating on Ω, we obtain 1𝑞0𝑑𝑑𝑡Ω||𝑤𝑛||𝑞0𝑞𝑑𝑥+0𝑐1Ω||𝑤𝑛||2||𝑤𝑛||𝑞02𝑑𝑥𝑙Ω||𝑤𝑛||𝑞0𝑑𝑥.(3.15) Since |𝑤𝑛|2|𝑤𝑛|𝑞02=(2/𝑞0)2|(|𝑤𝑛|(𝑞02)/2𝑤𝑛)|2, we deduce that 𝑑𝑑𝑡Ω||𝑤𝑛||𝑞0𝑑𝑥+Ω|||||𝑤𝑛||(𝑞02)/2𝑤𝑛|||2𝑑𝑥𝐶Ω||𝑤𝑛||𝑞0𝑑𝑥.(3.16) Integrating (3.16) between 𝑠 and 𝑡+1(𝑡𝑠<𝑡+1), it yields Ω||𝑤𝑛||(𝑡+1)𝑞0𝑑𝑥𝐶𝑠𝑡+1Ω||𝑤𝑛||𝑞0𝑑𝑥𝑑𝜉+Ω||𝑤𝑛||(𝑠)𝑞0𝑑𝑥.(3.17) Integrating the above inequality with respect to 𝑠 between 𝑡 and 𝑡+1, we get Ω||𝑤𝑛||(𝑡+1)𝑞0𝑑𝑥𝐶𝑡𝑡+1Ω||𝑤𝑛||𝑞0𝑑𝑥𝑑𝜉.(3.18) Therefore, Ω||𝑤𝑛(||𝑡)𝑞0𝑑𝑥𝐶,𝑡1.(3.19) Integrating (3.16) on [𝑡,𝑡+1] for 𝑡1, we deduce that 𝑡𝑡+1Ω||𝑤|𝑛||(𝑞02)/2𝑤𝑛|2𝑑𝑥𝑑𝜉𝐶𝑡𝑡+1Ω||𝑤𝑛||(𝜉)𝑞0𝑑𝑥𝑑𝜉+𝐶Ω||𝑤𝑛||(𝑡)𝑞0𝑑𝑥𝐶.(3.20) Note that (3.20) insures that, for any 𝑡1, there exists at least a 𝑡0[𝑡,𝑡+1] such that Ω|||||𝑤𝑛𝑡0||(𝑞02)/2𝑤𝑛𝑡0|||2𝑑𝑥𝐶.(3.21) Standard Sobolev imbedding implies that Ω||𝑤𝑛𝑡0||𝑞0(𝑁/(𝑁2))𝑑𝑥𝐶.(3.22) Now multiplying (3.5) by |𝑤𝑛|𝑞12𝑤𝑛,𝑞1=(𝑁/(𝑁2))𝑞0, we have 𝑁2𝑁𝑞0𝑑𝑑𝑡Ω||𝑤𝑛||(𝑁/(𝑁2))𝑞0𝑑𝑥+𝐶𝑐,𝑞0,𝑁Ω|||||𝑤𝑛||(𝑁𝑞02𝑁+4)/(2𝑁4)𝑤𝑛|||2𝑑𝑥𝑙Ω||𝑤𝑛||(𝑁/(𝑁2))𝑞0𝑑𝑥.(3.23) Using Hölder inequality, and Young inequality we deduce that Ω|𝑤𝑛|(𝑁/(𝑁2))𝑞0𝑑𝑥𝜀Ω|||||𝑤𝑛||(𝑁𝑞02𝑁+4)/(2𝑁4)𝑤𝑛|||2𝑑𝑥+𝐶𝜀Ω||𝑤𝑛||𝑞0𝑑𝑥𝑁/(𝑁2).(3.24) Taking (3.24) into (3.23), it yields 𝑑𝑑𝑡Ω||𝑤𝑛||(𝑁/(𝑁2))𝑞0𝑑x+𝐶𝑐,𝑞0,𝑁Ω|||||𝑤𝑛||(𝑁𝑞02𝑁+4)/(2𝑁4)𝑤𝑛|||2𝑑𝑥𝐶𝜀Ω||𝑤𝑛||𝑞0𝑑𝑥𝑁/(𝑁2).(3.25) Integrating (3.25) between 𝑡0 and 𝑡0+𝑠,0<𝑠1, we have Ω||𝑤𝑛𝑡0||+𝑠(𝑁/(𝑁2))𝑞0𝑑𝑥Ω||𝑤𝑛𝑡0||(𝑁/(𝑁2))𝑞0𝑑𝑥+𝐶𝜀Ω||𝑤𝑛||𝑞0𝑑𝑥𝑁/(𝑁2).(3.26) Therefore, from (3.19) and (3.22) we get Ω||𝑤𝑛(||𝑡)(𝑁/(𝑁2))𝑞0𝑑𝑥=Ω||𝑤𝑛(||𝑡)𝑞1𝑑𝑥𝐶,𝑡2,(3.27) with 𝐶 independent of 𝑛. Integrating (3.25) between 𝑡 and 𝑡+1 for 𝑡2, we obtain 𝑡𝑡+1Ω|||||𝑤𝑛||(𝑞12)/2𝑤𝑛|||2𝑑𝑥𝑑𝜉=𝑡𝑡+1Ω||𝑤|(𝑛||(𝑁𝑞02𝑁+4)/(2𝑁4)𝑤𝑛)|2𝑑𝑥𝑑𝜉𝐶.(3.28) Similar to (3.22), for any 𝑡2, there exists at least a 𝑡0[𝑡,𝑡+1] such that Ω|𝑤𝑛(𝑡0)|𝑞0(𝑁2/(𝑁2)2)𝑑𝑥𝐶.(3.29) Bootstrap the above processes, we can deduce that Ω||𝑤𝑛(||𝑡)𝑞𝑘𝑑𝑥𝐶,𝑡𝑇𝑘,(3.30) with 𝑞𝑘=(𝑁/(𝑁2))𝑘𝑞0, and 𝐶 independent of 𝑛. Note that 𝑤𝑛𝑤 in 𝐶([0,𝑇];𝐿1(Ω)) and 𝑤𝐶([0,𝑇];𝐿1(Ω)). From Lemma 2.6, we have 𝑤(𝑡)𝑞𝑘𝐿𝑞𝑘(Ω)=Ω||||𝑤(𝑡)𝑞𝑘𝑑𝑥𝐶,𝑡𝑇𝑘.(3.31) Taking 𝑘 large enough, we get the second part of (i) proved. If the integration are taken over [𝑡,𝑡+𝛿0] instead of [𝑡,𝑡+1], we get the first part of (i).
Now we are in the position to prove (ii). We multiply (3.5) with 𝑤𝑛 and deduce that 12𝑑𝑑𝑡Ω||𝑤𝑛||2𝑑𝑥+𝑐Ω||𝑤𝑛||2𝑑𝑥𝑙Ω||𝑤𝑛||2𝑑𝑥,(3.32) integrating over [𝑡,𝑡+1],𝑡𝑇, we get 𝑡𝑡+1Ω||𝑤𝑛||2𝑑𝑥𝑑𝑡𝐶,(3.33) with 𝐶 independent of 𝑛. Now, multiplying (3.5) with 𝑤𝑛𝑡, we obtain Ω||𝑤𝑛𝑡||2𝑑𝑑𝑥+𝑑𝑡Ω||𝑎(𝑥)𝑤𝑛||2𝑑𝑑𝑥+𝑑𝑡Ω(𝐹(𝑤𝑛+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛)𝑑𝑥=0,(3.34) where 𝐹(𝑣+𝜎)=𝜎0𝑓(𝑣+𝑠)𝑑𝑠. Integrating (3.34) between 𝑠 and 𝑡+1(𝑡𝑠<𝑡+1) gives Ω||𝑤𝑛(||𝑡+1)2𝑑𝑥+Ω(𝐹(𝑤𝑛(𝑡+1)+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛(𝑡+1))𝑑𝑥𝐶Ω||𝑤𝑛||(𝑠)2𝑑𝑥+𝐶Ω(𝐹(𝑤𝑛(𝑠)+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛(𝑠))𝑑𝑥.(3.35) Now, integrating the above inequality with respect to 𝑠 between 𝑡 and 𝑡+1 we have Ω||𝑤𝑛(||𝑡+1)2𝑑𝑥+Ω(𝐹(𝑤𝑛(𝑡+1)+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛(𝑡+1))𝑑𝑥𝐶𝑡𝑡+1Ω||𝑤𝑛||2𝑑𝑥𝑑𝜉+𝐶𝑡𝑡+1||||Ω(𝐹(𝑤𝑛+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛||||)𝑑𝑥𝑑𝜉.(3.36) Since Ω(𝐹(𝑤𝑛+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛)𝑑𝑥=Ω10(𝑓(𝑣𝑛+𝑠𝑤𝑛)𝑓(𝑣𝑛))𝑤𝑛𝑓𝑑𝑠𝑑𝑥,(𝑣𝑛+𝜏𝑤𝑛)𝑓(𝑣𝑛)=10𝑓(𝑣𝑛+𝜃𝜏𝑤𝑛)𝜏𝑤𝑛𝑑𝜃,0𝜏1.(3.37) We deduce that ||||Ω(𝐹(𝑤𝑛+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛||||)𝑑𝑥Ω||𝑓𝑣𝑛+𝜏𝑤𝑛||||𝑤𝑛||2𝑑𝑥,0𝜏1.(3.38) From the assumption (1.4) on 𝑓, we have ||||Ω(𝐹(𝑤𝑛+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛||||)𝑑𝑥Ω𝐶||𝑣𝑛||𝑝2+||𝑤𝑛||𝑝2||𝑤+1𝑛||2𝑑𝑥𝐶𝑣𝑛𝐿𝑝2𝑝1(Ω)+𝑤𝑛2𝐿2𝑝2(Ω)+𝑤𝑛𝑝𝐿𝑝(Ω).+1(3.39) Using results in (3.13) and (3.30), we know that ||||Ω(𝐹(𝑤𝑛+𝑣𝑛)𝐹(𝑣𝑛)𝑓(𝑣𝑛)𝑤𝑛||||)𝑑𝑥𝐶,𝑡𝑇.(3.40) Set 𝑇0=max{𝑇,𝑇}. Combining (3.33), (3.34), (3.36), and (3.40) we have Ω||𝑤𝑛(||𝑡)2𝑑𝑥𝐶,𝑡𝑇0+1,(3.41)𝑡𝑡+1Ω||𝑤𝑛𝑡||2𝑑𝑥𝑑𝜉𝐶,𝑡𝑇0+1.(3.42) From Lemma 2.6, we obtain 𝑤𝐻10(Ω)𝐶,𝑡𝑇0+1.(3.43) Thus we get the second part of (ii) proved. Taking integration over [𝑡,𝑡+𝛿0] instead of [𝑡,𝑡+1], the first part of (ii) follows. The proof is completed now.

4. Proof of the Main Theorems

Let {𝑆(𝑡)}𝑡0 be the semigroup generated by problem (1.1) and let 𝑣(𝑥) be an approximated solution to problem (2.3). Define 𝑆1𝑢(𝑡)0𝑣(𝑥)=𝑆(𝑡)𝑢0𝑣(𝑥).(4.1) Then it is easy to verify that {𝑆1(𝑡)}𝑡0 is a continuous semigroup in (𝐿1(Ω)𝑣) and hence in 𝐿1(Ω). From the results in Section 3, we know that the semigroup {𝑆(𝑡)}𝑡0 possesses a global attractor 𝒜 in 𝐿1(Ω). To verify the second part of Theorem 1.1, we prove the following theorem.

Theorem 4.1. Under the assumptions of Theorem 1.1, the semigroup {𝑆1(𝑡)}𝑡0 possesses a global attractor 𝒜𝑣 in 𝐿𝑟(Ω)𝐻10(Ω),1𝑟<, that is, 𝒜𝑣 is compact, invariant in 𝐿𝑟(Ω)𝐻10(Ω),1𝑟<, and attracts every bounded initial set of 𝐿1(Ω) in the norm topology of 𝐿𝑟(Ω)𝐻10(Ω),1𝑟<.

Proof. From Theorem 3.1, we know that {𝑆(𝑡)}𝑡0 possesses an absorbing set 𝐵0 in 𝐿1(Ω). Also {𝑆1(𝑡)}𝑡0 possesses absorbing sets 𝐵1(=𝐵0𝑣), 𝐵2, respectively, in 𝐿1(Ω) and 𝐿𝑟(Ω)𝐻10(Ω) for any 1𝑟<.
In the next, we prove the asymptotic compactness of {𝑆1(𝑡)}𝑡0. Before that we establish the following estimate Ω||𝑤𝑛𝑡||2𝑑𝑥𝐶,for𝑡largeenough.(4.2) Actually, differentiating (3.5) in time and denoting 𝑤𝑛=𝑤𝑛𝑡, we have 𝑤𝑛𝑡𝑤div𝑎(𝑥)𝑛+𝑓(𝑣𝑛+𝑤𝑛)𝑤𝑛=0.(4.3) Multiplying (4.3) by 𝑤𝑛 and using (1.3), we deduce that 𝑑𝑑𝑡Ω||𝑤𝑛||2𝑑𝑥𝑙Ω||𝑤𝑛||2𝑑𝑥.(4.4) Integrating the above inequality between 𝑠 and 𝑡+1(𝑡𝑠<𝑡+1) gives Ω||𝑤𝑛||(𝑡+1)2𝑑𝑥2𝑙𝑠𝑡+1Ω||𝑤𝑛||2𝑑𝑥𝑑𝜉+Ω||𝑤𝑛||(𝑠)2𝑑𝑥.(4.5) Integrating the above inequality with respect to 𝑠 between 𝑡 and 𝑡+1, using (3.42) we obtain Ω||𝑤𝑛||(𝑡)2𝑑𝑥(2𝑙+1)𝑡𝑡+1Ω||𝑤𝑛||2𝑑𝑥𝑑𝜉(2𝑙+1)𝐶,𝑡𝑇0+1.(4.6) Now we prove the asymptotic compactness of the semigroup {𝑆1(𝑡)}𝑡0, that is, for any sequences {𝑤0,𝑘}𝐵1,𝑡𝑘, sequence {𝑤𝑘(𝑡𝑘)} has convergent subsequences, where 𝑤𝑘(𝑡)=𝑆1(𝑡)𝑤0,𝑘. Since {𝑆1(𝑡)}𝑡0 is compact in 𝐿𝑟(Ω),1𝑟<, there is a subsequence {𝑤𝑘𝑖(𝑡𝑘𝑖)}, which is a Cauchy sequence in 𝐿𝑟(Ω),1𝑟<. Denoting 𝑢𝑘𝑖(𝑡𝑘𝑖)=𝑤𝑘𝑖(𝑡𝑘𝑖)+𝑣,𝑢𝑘𝑗(𝑡𝑘𝑗)=𝑤𝑘𝑗(𝑡𝑘𝑗)+𝑣, we deduce that 𝑐𝑤𝑘𝑖(𝑡𝑘𝑖)𝑤𝑘𝑗𝑡𝑘𝑗2𝐻10(Ω)𝑤div𝑎(𝑥)𝑘𝑖𝑡𝑘𝑖𝑤+𝑣div𝑎(𝑥)𝑘𝑗𝑡𝑘𝑗+𝑣,𝑤𝑘𝑖𝑡𝑘𝑖𝑤𝑘𝑗𝑡𝑘𝑗=𝜕𝑡𝑤𝑘𝑖𝑡𝑘𝑖𝜕𝑡𝑤𝑘𝑗𝑡𝑘𝑗𝑢+𝑓𝑘𝑖𝑡𝑘𝑖𝑢𝑓𝑘𝑗𝑡𝑘𝑗,𝑤𝑘𝑖𝑡𝑘𝑖𝑤𝑘𝑗𝑡𝑘𝑗𝜕𝑡𝑤𝑘𝑖(𝑡𝑘𝑖)𝜕𝑡𝑤𝑘𝑗(𝑡𝑘𝑗)𝐿2(Ω)𝑤𝑘𝑖(𝑡𝑘𝑖)𝑤𝑘𝑗𝑡𝑘𝑗𝐿2(Ω)+𝑓𝑤𝑘𝑖𝑡𝑘𝑖𝑤+𝑣𝑓𝑘𝑗𝑡𝑘𝑗+𝑣𝐿𝜎(Ω)𝑤𝑘𝑖𝑡𝑘𝑖𝑤𝑘𝑗𝑡𝑘𝑗𝐿𝜎(Ω).(4.7) We then conclude form (4.6) that {𝑤𝑘𝑖(𝑡𝑘𝑖)} is a Cauchy sequence in 𝐻10(Ω), and thus {𝑆1(𝑡)}𝑡0 is asymptotically compact in 𝐿𝑟(Ω)𝐻10(Ω),1𝑟<.
Using Lemma 2.7, we conclude that {𝑆1(𝑡)}𝑡0 possesses a global attractor 𝒜𝑣, which is compact, invariant in 𝐿𝑟(Ω)𝐻10(Ω), and attracts every bounded initial sets of 𝐿1(Ω) in the topology of 𝐿𝑟(Ω)𝐻10(Ω).

Completion of the Proof of Theorem 1.1
Note that 𝒜𝑣=𝑠0𝑡𝑠𝑆1(𝑡)𝐵2𝐿1(Ω)=𝑠0𝑡𝑠𝑆1(𝑡)𝐵1𝐿1(Ω)=𝑠0𝑡𝑠𝑆(𝑡)𝐵0𝑣𝐿1(Ω),𝒜=𝑠0𝑡𝑠𝑆(𝑡)𝐵0𝐿1(Ω).(4.8) Thus we have 𝒜=𝒜𝑣+𝑣.(4.9) The above relation between 𝒜 and 𝒜𝑣 implies the conclusion of Theorem 1.1 directly.

Proof of Theorem 1.2. Let 𝑢(𝑡),𝑣 be the approximated solution to the parabolic and its corresponding elliptic problem respectively. Since the approximated solution is a duality solution and conversely, we conclude that 𝑢(𝑡) converges to 𝑣 in 𝐿1(Ω) as 𝑡. Using arguments similar to Section 3, we can prove similar regularity results for 𝑤(=𝑢𝑣) and then prove the asymptotic compactness of the semigroup 𝑆1(𝑡) as in Theorem 4.1. Thus, we obtain that 𝑤(𝑡) converges to 0 in 𝐿𝑟(Ω),1𝑟<, as 𝑡. Moreover from the asymptotic compactness of the semigroup 𝑆1(𝑡), we know that 𝑤(𝑡) converges to 0 in 𝐻10(Ω) as 𝑡. Else, we have a sequence 𝑡𝑛, such that 𝐶>𝑤(𝑡𝑛)𝐻10(Ω)𝜖>0. Since the semigroup 𝑆1(𝑡) is asymptotically compact, there is a subsequence 𝑡𝑛𝑗, such that 𝑤(𝑡𝑛𝑗) converges to a function 𝜒 in 𝐻10(Ω) and hence in 𝐿1(Ω). Thus 𝜒=0. A contradiction!

Acknowledgments

The authors want to thank the referee for the careful reading of the manuscript. This work is partially supported by the 211 Project of Anhui University (KJTD002B/32030015) and the NSFC Grant (11071001, 11271019).

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