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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 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.Β Zelik

Copyright Β© 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 Ξ© be a smooth bounded domain in ℝ𝑁,(𝑁β‰₯3). We consider the asymptotic behavior of solutions to the following problem π‘’π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’)+πœ†π‘“(𝑒)=πœ‡inΩ×ℝ+,𝑒=0onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=𝑒0(π‘₯)inΞ©, where 𝑒0∈𝐿1(Ξ©), πœ‡ 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 π‘’π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’)+πœ†π‘“(𝑒)=πœ‡,inΩ×ℝ+,𝑒=0,onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=𝑒0(π‘₯),inΞ©,(1.1) where Ξ© is a bounded domain in ℝ𝑁(𝑁β‰₯2) with smooth boundary πœ•Ξ©, 𝑒0∈𝐿1(Ξ©), πœ†β‰₯0,πœ‡ is a finite Radon measure independent of time, π‘Ž(π‘₯) is a matrix with bounded, measurable entries, and satisfying the ellipticity assumption ||πœ‰||π‘Ž(π‘₯)πœ‰β‹…πœ‰β‰₯𝑐2,foranyπœ‰βˆˆβ„N,with𝑐>0.(1.2) Concerning the nonlinear term, we assume that 𝑓 is a 𝐢1 function satisfying, π‘“ξ…ž||𝑓(𝑠)β‰₯βˆ’π‘™,(1.3)ξ…ž||ξ€·(𝑠)≀𝐢1+|𝑠|π‘βˆ’2ξ€Έ,𝑝β‰₯2,(1.4)𝐢|𝑠|π‘βˆ’π‘˜β‰€π‘“(𝑠)𝑠≀𝐢|𝑠|𝑝+π‘˜,𝑝β‰₯2,(1.5) both for all π‘ βˆˆπ‘…, where 𝑙,𝐢,π‘˜ 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 [1–4]. Existence and regularity results for parabolic equations with 𝐿1 and measure data have been studied widely by many authors in the past decades, see [5–10]. 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 π‘’βˆˆπΏ1((0,𝑇);π‘Š01,1(Ξ©)) such that 𝑓(𝑒)∈𝐿1(𝑄𝑇) for any 𝑇>0, and βˆ’ξ€œπ‘„π‘‡π‘’πœ‘π‘‘ξ€œπ‘‘π‘₯𝑑𝑑+π‘„π‘‡ξ€œπ‘Ž(π‘₯)βˆ‡π‘’βˆ‡πœ‘π‘‘π‘₯𝑑𝑑+πœ†π‘„π‘‡ξ€œπ‘“(𝑒)πœ‘π‘‘π‘₯𝑑𝑑=π‘„π‘‡ξ€œπœ‘π‘‘πœ‡+Ω𝑒0(π‘₯)πœ‘(0,π‘₯)𝑑π‘₯,(1.6) for any πœ‘βˆˆπΆβˆžπ‘([0,𝑇)Γ—Ξ©).

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 β€œbetter” 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 [11–16]. 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 πœ†>0, to ensure the existence result for large 𝑝 in (1.5) [17], we restrict ourselves to diffuse measures, that is, πœ‡ 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 π’œ in πΏπ‘βˆ’1(Ξ©)βˆ©π‘Š01,π‘ž(Ξ©) with π‘ž<max{𝑁/(π‘βˆ’1),(2π‘βˆ’2)/𝑝}, which attracts every bounded subset of 𝐿1(Ξ©) in the norm of πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©), for any π‘Ÿβˆˆ[1,∞).

For the case πœ†=0, we consider general bounded Radon measure πœ‡ 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 πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©), for any π‘Ÿβˆˆ[1,∞), though they all lie in some less regular spaces.

Our main results can be stated as follows.

Theorem 1.1. Assume that 𝑒0∈𝐿1(Ξ©), πœ†>0, πœ‡ is a bounded Radon measure, which does not charge the sets of zero parabolic 2-capacity and is independent of time, 𝑓 is a 𝐢1 function satisfying assumptions (1.3)–(1.5). Then the semigroup {𝑆(𝑑)}𝑑β‰₯0, generated by approximated solutions of problem (1.1), possesses a global attractor π’œ in 𝐿1(Ξ©). Moreover, π’œ is compact and invariant in πΏπ‘βˆ’1(Ξ©)βˆ©π‘Š01,π‘ž(Ξ©) with π‘ž<max{𝑁/(π‘βˆ’1),(2π‘βˆ’2)/𝑝}, and attracts every bounded subset of 𝐿1(Ξ©) in the norm topology of πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©), 1β‰€π‘Ÿ<∞.

Theorem 1.2. Assume that 𝑒0∈𝐿1(Ξ©), πœ†=0, πœ‡ is a bounded Radon measure independent of time. Then the approximated solution 𝑒(𝑑) of problem (1.1) is unique and converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©), for any 1β‰€π‘Ÿ<∞.

Remark 1.3. Though 𝑒(𝑑) and 𝑣 all lie in some less-regular spaces, 𝑒(𝑑) converges to 𝑣 in stronger norm, that is, 𝑒(𝑑)βˆ’π‘£ converges to 0 in πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©), 1β‰€π‘Ÿ<∞. Such a result, in some sense, sharpens the result of [13], where the author showed that 𝑒(𝑑) converges to 𝑣 in 𝐿1(Ξ©).

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 𝐢 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 𝑒 is called an approximated solution of problem (1.1), if π‘’βˆˆπΏ1((0,𝑇);π‘Š01,1(Ξ©)), 𝑓(𝑒)∈𝐿1(𝑄𝑇) for any 𝑇>0, and βˆ’ξ€œπ‘„π‘‡π‘’πœ‘π‘‘ξ€œπ‘‘π‘₯𝑑𝑑+π‘„π‘‡ξ€œπ‘Ž(π‘₯)βˆ‡π‘’βˆ‡πœ‘π‘‘π‘₯𝑑𝑑+πœ†π‘„π‘‡ξ€œπ‘“(𝑒)πœ‘π‘‘π‘₯𝑑𝑑=π‘„π‘‡ξ€œπœ‘π‘‘πœ‡+Ω𝑒0(π‘₯)πœ‘(0,π‘₯)𝑑π‘₯,(2.1) for any πœ‘βˆˆπΆβˆžπ‘([0,𝑇)Γ—Ξ©), and moreover, 𝑒 is obtained as limit of solutions to the following approximated problem π‘’π‘›π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’π‘›)+πœ†π‘“(𝑒𝑛)=πœ‡π‘›,inΩ×ℝ+,𝑒=0,onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=𝑒𝑛0,inΞ©,(2.2) where {πœ‡π‘›},{𝑒𝑛0} is a smooth approximation of data πœ‡,𝑒0.

Theorem 2.2. Under the assumptions of Theorem 1.1, problem (1.1) has a unique approximated solution π‘’βˆˆπΆ([0,𝑇];𝐿1(Ξ©)), for any 𝑇>0, satisfying (i) if 𝑝β‰₯(2𝑁+2)/𝑁, then π‘’βˆˆπΏπ‘ž((0,𝑇);π‘Š01,π‘ž(Ξ©)) with π‘ž<(2π‘βˆ’2)/𝑝;(ii) if 2≀𝑝<(2𝑁+2)/𝑁, then π‘’βˆˆπΏπ‘ž((0,𝑇);π‘Š01,π‘ž(Ξ©)) with π‘ž<(𝑁+2)/(𝑁+1).

Proof. According to [18, Theorem 2.12], if a Radon measure πœ‡ on 𝑄𝑇 does not charge the sets of zero parabolic 2-capacity and is independent of time, πœ‡ 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], πœ‡ can be decomposed as πœ‡=𝑔+div𝐺, where π‘”βˆˆπΏ1(Ξ©),𝐺∈(𝐿2(Ξ©))𝑁. Hence, we need only to consider the following problem π‘’π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’)+πœ†π‘“(𝑒)=𝑔+div𝐺,inΩ×ℝ+,𝑒=0,onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=𝑒0,inΞ©.(2.3) The proof of existence part of the theorem is similar to [9]. Besides, one can prove π‘’βˆˆπΆ([0,𝑇];𝐿1(Ξ©)) using arguments similar to CLAIM 2 in [8]. So we omit the details of them and only prove the uniqueness result.
Let {𝑔𝑛}π‘›βˆˆβ„•, {𝑒𝑛0}π‘›βˆˆβ„• be a smooth approximation of data 𝑔 and 𝑒0 with ‖‖𝑒𝑛0‖‖𝐿1(Ξ©)≀‖‖𝑒0‖‖𝐿1(Ξ©),‖̃𝑔𝑛‖𝐿1(Ξ©)≀‖𝑔‖𝐿1(Ξ©),(2.4) and let {̃𝑔𝑛}π‘›βˆˆβ„•, {̃𝑒𝑛0}π‘›βˆˆβ„• be another smooth approximation of the data with ‖‖̃𝑒𝑛0‖‖𝐿1(Ξ©)≀‖‖𝑒0‖‖𝐿1(Ξ©),‖̃𝑔𝑛‖𝐿1(Ξ©)≀‖𝑔‖𝐿1(Ξ©).(2.5) Assume that 𝑒,̃𝑒 are two approximated solutions to problem (1.1), obtained as limit of the solutions to the following two approximated problems, respectively, π‘’π‘›π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’π‘›)+πœ†π‘“(𝑒𝑛)=𝑔𝑛+div𝐺,inΩ×ℝ+,𝑒=0,onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=𝑒𝑛0,inΞ©,(2.6)Μƒπ‘’π‘›π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡Μƒπ‘’π‘›)+πœ†π‘“(̃𝑒𝑛)=̃𝑔𝑛+div𝐺,inΩ×ℝ+,𝑒=0,onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=̃𝑒𝑛0,inΞ©.(2.7) Now we prove that 𝑒=̃𝑒. For any π‘˜>0, define πœ“π‘˜(𝑠) as πœ“π‘˜(⎧βŽͺ⎨βŽͺβŽ©π‘ )=π‘˜,𝑠>π‘˜π‘ ,|𝑠|β‰€π‘˜βˆ’π‘˜,𝑠<βˆ’π‘˜.(2.8) Let Ξ¨π‘˜βˆ«(𝜎)=𝜎0πœ“π‘˜(𝑠)𝑑𝑠 be its primitive function. Taking πœ“π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›) as a test function in (2.6) and (2.7), we deduce that ξ€œΞ©Ξ¨π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›ξ€œ)(𝑑)𝑑π‘₯βˆ’Ξ©Ξ¨π‘˜ξ€·π‘’π‘›0βˆ’Μƒπ‘’π‘›0ξ€Έξ€œπ‘‘π‘₯+π‘„π‘‡ξ€·π‘Ž(π‘₯)βˆ‡πœ“π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›)ξ€Έ2ξ€œπ‘‘π‘₯𝑑𝑑=πœ†π‘„π‘‡(𝑓(̃𝑒𝑛)βˆ’π‘“(𝑒𝑛))πœ“π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›ξ€œ)𝑑π‘₯𝑑𝑑+𝑄𝑇(π‘”π‘›βˆ’Μƒπ‘”π‘›)πœ“π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›β‰€ξ€œ)𝑑π‘₯π‘‘π‘‘π‘„π‘‡πœ†π‘™(π‘’π‘›βˆ’Μƒπ‘’π‘›)πœ“π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›ξ€œ)𝑑π‘₯𝑑𝑑+𝑄𝑇||π‘”π‘›βˆ’Μƒπ‘”π‘›||𝑑π‘₯𝑑𝑑.(2.9) Hence, from the assumptions on 𝑓, we get ξ€œΞ©Ξ¨π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›ξ€œ)(𝑑)𝑑π‘₯≀2π‘™πœ†π‘‡0ξ€œΞ©Ξ¨π‘˜(π‘’π‘›βˆ’Μƒπ‘’π‘›ξ€œ)𝑑π‘₯𝑑𝑑+𝑄𝑇||π‘”π‘›βˆ’Μƒπ‘”π‘›||ξ€œπ‘‘π‘₯𝑑𝑑+Ξ©Ξ¨π‘˜ξ€·π‘’π‘›0βˆ’Μƒπ‘’π‘›0𝑑π‘₯.(2.10) Let π‘›β†’βˆž, we have ξ€œΞ©Ξ¨π‘˜ξ€œ(π‘’βˆ’Μƒπ‘’)(𝑑)𝑑π‘₯≀2π‘™πœ†π‘‡0ξ€œΞ©Ξ¨π‘˜(π‘’βˆ’Μƒπ‘’)(𝑑)𝑑π‘₯𝑑𝑑.(2.11) Thus for all π‘˜>0, we have sup[]0,π‘‡ξ€œΞ©Ξ¨π‘˜(Μƒπ‘’βˆ’π‘’)(𝑑)𝑑π‘₯≀2𝑙𝑇sup[]0,π‘‡ξ€œΞ©Ξ¨π‘˜(Μƒπ‘’βˆ’π‘’)𝑑π‘₯.(2.12) Taking 𝑇′ small enough such that 2π‘™πœ†π‘‡β€²<1, we deduce that Ξ¨π‘˜(π‘’βˆ’π‘’)=0 for all π‘˜>0 in π‘„π‘‡ξ…ž, thus 𝑒≑𝑒 in π‘„π‘‡ξ…ž. 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 𝑝β‰₯(2π‘βˆ’2)/(π‘βˆ’2), we have π‘£βˆˆπ‘Š01,π‘ž(Ξ©) for π‘ž<(2π‘βˆ’2)/𝑝;(ii)if 2≀𝑝<(2π‘βˆ’2)/(π‘βˆ’2), we have π‘£βˆˆπ‘Š01,π‘ž(Ξ©) for π‘ž<𝑁/(π‘βˆ’1).

Remark 2.4. Note that if 𝑣 is an approximated solution to following problem βˆ’div(π‘Ž(π‘₯)βˆ‡π‘£)+πœ†π‘“(𝑣)=𝑔+div𝐺,inΞ©,𝑣=0,onπœ•Ξ©,(2.13) then there is a sequence {𝑣𝑛} converges to 𝑣, where 𝑣𝑛 is the solution of the corresponding approximated problem βˆ’div(π‘Ž(π‘₯)βˆ‡π‘£π‘›)+πœ†π‘“(𝑣𝑛)=𝑔𝑛𝑣+div𝐺,inΞ©,𝑛=0,onπœ•Ξ©.(2.14) And hence 𝑣𝑛 is a solution of parabolic equations π‘£π‘›π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘£π‘›)+πœ†π‘“(𝑣𝑛)=𝑔𝑛+div𝐺,inΩ×ℝ+,𝑣𝑛=0,onπœ•Ξ©Γ—β„+,𝑣𝑛(0)=𝑣𝑛(π‘₯),inΞ©.(2.15) Thus, 𝑣 is an approximated solution of problem (1.1) with initial data 𝑒0=𝑣(π‘₯).

Under the assumptions of Theorem 1.2, the problem turns out to be π‘’π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’)=πœ‡,inΩ×ℝ+,𝑒=0,onπœ•Ξ©Γ—β„+,𝑒(π‘₯,0)=𝑒0,inΞ©.(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 𝑒 be an approximated solution, then there exist a smooth approximation {πœ‡π‘›}π‘›βˆˆβ„•, {𝑒𝑛0}π‘›βˆˆβ„• of data πœ‡ and 𝑒0, such that the solution of the approximated problem of (2.16) with data πœ‡π‘› and 𝑒𝑛0 converges to 𝑒. Let β„ŽβˆˆπΆβˆžπ‘(𝑄𝑇) and πœ” be the solution of the the following parabolic problem βˆ’πœ”π‘‘ξ€·π‘Žβˆ’divβˆ—ξ€Έ(π‘₯)βˆ‡πœ”=β„Ž,inΩ×(0,𝑇),πœ”=0,onπœ•Ξ©Γ—(0,𝑇),πœ”(π‘₯,𝑇)=0,inΞ©,(2.17) where π‘Žβˆ—(π‘₯) is the transposed matrix of π‘Ž(π‘₯). Taking πœ” as a test function in the approximated problem and taking 𝑒𝑛 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, π‘Œ be two Banach spaces, let 𝑋 be separable, reflexive, and let π‘‹βŠ‚π‘Œ with dual π‘‹βˆ—. Suppose that {𝑒𝑛} is uniformly bounded in 𝐿∞((0,𝑇);𝑋) with esssupπ‘‘βˆˆ[0,𝑇]‖𝑒𝑛(𝑑)‖𝑋≀𝐢,(2.18) and that 𝑒𝑛→𝑒 weakly in πΏπ‘Ÿ((0,𝑇);𝑋) for some π‘Ÿβˆˆ(1,∞). Then esssupπ‘‘βˆˆ[0,𝑇]‖𝑒(𝑑)‖𝑋≀𝐢.(2.19) Moreover, if π‘’βˆˆπΆ([0,𝑇];π‘Œ), then in fact supπ‘‘βˆˆ[0,𝑇]‖𝑒(𝑑)‖𝑋≀𝐢.(2.20)

Lemma 2.7 (see [16]). Let 𝑋,π‘Œ be two Banach spaces with imbedding 𝑋β†ͺπ‘Œ, let {𝑆(𝑑)}𝑑β‰₯0 be a continuous semigroup on π‘Œ. Assume that {𝑆(𝑑)}𝑑β‰₯0 is asymptotically compact in 𝑋 and has an absorbing set 𝐡0βŠ‚π‘‹, that is, for any bounded set π΅βŠ‚π‘Œ, there exists a 𝑇=𝑇(𝐡) such that 𝑆(𝑑)π΅βŠ‚π΅0,βˆ€π‘‘β‰₯𝑇.(2.21) Then {𝑆(𝑑)}𝑑β‰₯0 has a global attractor π’œ in 𝑋, which is compact, invariant in 𝑋 and attracts every bounded sets of π‘Œ 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 [23–27]. 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 π‘Œ.

3. Improved Regularity Results on the Approximated Solutions

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

Theorem 3.1. Under the assumptions of Theorem 2.2, let 𝑒(𝑑) be the approximated solution to problem (1.1). Then 𝑒 admits the decomposition 𝑒(π‘₯,𝑑)=𝑀(π‘₯,𝑑)+𝑣(π‘₯), with 𝑣 being an approximated solution to problem (2.13), and 𝑀 being an approximated solution of the following problem π‘€π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘€)+πœ†π‘“(𝑣+𝑀)βˆ’πœ†π‘“(𝑣)=0,inΩ×ℝ+,𝑀=0,onπœ•Ξ©Γ—β„+,𝑀(π‘₯,0)=𝑒0βˆ’π‘£,inΞ©.(3.1) Moreover, we have (i)π‘€βˆˆπΏβˆž((𝛿,𝑇);πΏπ‘ž(Ξ©)) for any 0<𝛿<𝑇,1β‰€π‘ž<∞. Moreover, there exists a constant π‘€π‘ž and a time π‘‘π‘ž(𝑒0,𝑔,𝐺) such that ‖𝑀(𝑑)β€–πΏπ‘ž(Ξ©)β‰€π‘€π‘žforall𝑑β‰₯π‘‘π‘ž(𝑒0,𝑔,𝐺).(ii)π‘€βˆˆπΏβˆž((𝛿,𝑇);𝐻10(Ξ©)) for any 0<𝛿<𝑇. Moreover, there exists a constant 𝜌 and a time 𝑇0(𝑒0,𝑔,𝐺) such that ‖𝑀(𝑑)‖𝐻10(Ξ©)β‰€πœŒforall𝑑β‰₯𝑇0(𝑒0,𝑔,𝐺).

Proof. We follow the lines of [15, 28]. Let {𝑔𝑛} be a sequence of smooth data which converges to 𝑔 in 𝐿1(Ξ©) and ‖𝑔𝑛‖𝐿1(Ξ©)≀‖𝑔‖𝐿1(Ξ©). Let 𝑣𝑛 be a solution of the following approximated problem for each 𝑛, βˆ’div(π‘Ž(π‘₯)βˆ‡π‘£π‘›)+πœ†π‘“(𝑣𝑛)=𝑔𝑛𝑣+div𝐺,inΞ©,𝑛=0,onπœ•Ξ©.(3.2) Then 𝑣𝑛 converges (up to subsequences) to an approximated solution 𝑣 strongly in 𝐿1(Ξ©), and weakly in π‘Š01,π‘ž(Ξ©),1β‰€π‘ž<𝑁/(π‘βˆ’1). Let {𝑒𝑛} be a sequence of solutions to the following approximated problem π‘’π‘›π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘’π‘›)+πœ†π‘“(𝑒𝑛)=𝑔𝑛(π‘₯)+div𝐺,inΩ×ℝ+,𝑒𝑛(π‘₯)=0,onπœ•Ξ©Γ—β„+,𝑒𝑛(π‘₯,0)=𝑒𝑛0,inΞ©,(3.3) where 𝑒𝑛0 converges to 𝑒0 with ‖𝑒𝑛0‖𝐿1(Ξ©)≀‖𝑒0‖𝐿1(Ξ©). Similar to [8, 29], we know that 𝑒𝑛→𝑒weaklyinπΏπ‘žξ‚€(0,𝑇);π‘Š01,π‘žξ‚(Ξ©),π‘ž<𝑁+2,𝑒𝑁+1𝑛[]βŸΆπ‘’in𝐢0,𝑇;𝐿1ξ€Έ.(Ξ©)(3.4) Now let 𝑀𝑛(𝑑)=𝑒𝑛(𝑑)βˆ’π‘£π‘›. Then 𝑀𝑛 satisfies π‘€π‘›π‘‘βˆ’div(π‘Ž(π‘₯)βˆ‡π‘€π‘›)+πœ†π‘“(𝑣𝑛+𝑀𝑛)βˆ’πœ†π‘“(𝑣𝑛)=0,inΩ×ℝ+,𝑀𝑛=0,onπœ•Ξ©Γ—β„+,𝑀𝑛(π‘₯,0)=𝑒𝑛0βˆ’π‘£π‘›,inΞ©.(3.5) Similarly, we have 𝑀𝑛 (up to subsequences) converges to the approximated solution 𝑀 of problem (3.1) in 𝐢([0,𝑇];𝐿1(Ξ©)) and weakly in πΏπ‘ž((0,𝑇);π‘Š01,π‘ž(Ξ©)),π‘ž<(𝑁+2)/(𝑁+1).
Now we prove (i). Taking πœ“1(𝑒𝑛) as test function in (3.3) (for simplicity we take πœ†=1), we deduce that π‘‘ξ€œπ‘‘π‘‘Ξ©Ξ¨1(𝑒𝑛)ξ€œπ‘‘π‘₯+Ω𝑓(𝑒𝑛)πœ“1(𝑒𝑛)𝑑π‘₯≀‖𝑔‖𝐿1(Ξ©)+𝐢‖𝐺‖𝐿2(Ξ©).(3.6) Since 𝑓(𝑒𝑛)πœ“1(𝑒𝑛𝐢||𝑒)β‰₯𝑛||π‘βˆ’1||πœ“βˆ’πΆ1(𝑒𝑛)||,(3.7) we have π‘‘ξ€œπ‘‘π‘‘Ξ©Ξ¨1(𝑒𝑛)ξ€œπ‘‘π‘₯+𝐢Ω||Ξ¨1(𝑒𝑛)||𝑑π‘₯≀‖𝑔‖𝐿1(Ξ©)+𝐢‖𝐺‖𝐿2(Ξ©)||Ξ©||.+𝐢(3.8) The Gronwall's inequality implies that ξ€œΞ©Ξ¨1(𝑒𝑛(‖‖𝑒𝑑))𝑑π‘₯≀0‖‖𝐿1(Ξ©)π‘’βˆ’πΆπ‘‘||Ξ©||+𝐢+𝐢‖𝑔‖𝐿1(Ξ©)+𝐢‖𝐺‖𝐿2(Ξ©).(3.9) Noticing that ξ€œΞ©||𝑒𝑛(||ξ€œπ‘‘)𝑑π‘₯≀ΩΨ1(𝑒𝑛(||Ξ©||,𝑑))𝑑π‘₯+(3.10) we obtain that ξ€œΞ©||𝑒𝑛(||‖‖𝑒𝑑)𝑑π‘₯≀0‖‖𝐿1(Ξ©)π‘’βˆ’πΆπ‘‘||Ξ©||+𝐢+𝐢‖𝑔‖𝐿1(Ξ©)+𝐢‖𝐺‖𝐿2(Ξ©),𝑑β‰₯0.(3.11) Moreover, integrating (3.6) between 𝑑 and 𝑑+1 and using (3.7) we have ξ€œπ‘‘π‘‘+1ξ€œΞ©||𝑒𝑛||π‘βˆ’1‖‖𝑒𝑑π‘₯π‘‘πœ‰β‰€πΆ0‖‖𝐿1(Ξ©)π‘’βˆ’πΆπ‘‘+||Ξ©||+‖𝑔‖𝐿1(Ξ©)+‖𝐺‖𝐿2(Ξ©)ξ€Έ.(3.12) Similarly, taking πœ“1(𝑣𝑛) as test function in (3.2), we can deduce that ξ€œΞ©||𝑓(𝑣𝑛)||ξ€œπ‘‘π‘₯≀𝐢Ω||𝑣𝑛||π‘βˆ’1+1𝑑π‘₯≀𝐢‖𝑔‖𝐿1(Ξ©)+‖𝐺‖𝐿2(Ξ©)+||Ξ©||ξ€Έ.(3.13) Hence, ξ€œπ‘‘π‘‘+1ξ€œΞ©||𝑀𝑛||π‘βˆ’1𝑑π‘₯π‘‘πœ‰β‰€πΆβ€–π‘”β€–πΏ1(Ξ©)+‖𝐺‖𝐿2(Ξ©)+‖‖𝑒0‖‖𝐿1(Ξ©)+||Ξ©||ξ€Έ,(3.14) with 𝐢 independent of 𝑛, for 𝑑β‰₯0.
Now we use bootstrap method in the case 𝑝β‰₯3. The case 2≀𝑝<3 can be treated similarly with minor modifications. Multiplying (3.5) by |𝑀𝑛|π‘ž0βˆ’2𝑀𝑛,π‘ž0=π‘βˆ’1β‰₯2, and integrating on Ξ©, we obtain 1π‘ž0π‘‘ξ€œπ‘‘π‘‘Ξ©||𝑀𝑛||π‘ž0ξ€·π‘žπ‘‘π‘₯+0ξ€Έπ‘ξ€œβˆ’1Ξ©||βˆ‡π‘€π‘›||2||𝑀𝑛||π‘ž0βˆ’2ξ€œπ‘‘π‘₯≀𝑙Ω||𝑀𝑛||π‘ž0𝑑π‘₯.(3.15) Since |βˆ‡π‘€π‘›|2|𝑀𝑛|π‘ž0βˆ’2=(2/π‘ž0)2|βˆ‡(|𝑀𝑛|(π‘ž0βˆ’2)/2𝑀𝑛)|2, we deduce that π‘‘ξ€œπ‘‘π‘‘Ξ©||𝑀𝑛||π‘ž0ξ€œπ‘‘π‘₯+Ξ©|||βˆ‡ξ‚€||𝑀𝑛||(π‘ž0βˆ’2)/2𝑀𝑛|||2ξ€œπ‘‘π‘₯≀𝐢Ω||𝑀𝑛||π‘ž0𝑑π‘₯.(3.16) Integrating (3.16) between 𝑠 and 𝑑+1(𝑑≀𝑠<𝑑+1), it yields ξ€œΞ©||𝑀𝑛||(𝑑+1)π‘ž0ξ€œπ‘‘π‘₯≀𝐢𝑠𝑑+1ξ€œΞ©||𝑀𝑛||π‘ž0ξ€œπ‘‘π‘₯π‘‘πœ‰+Ξ©||𝑀𝑛||(𝑠)π‘ž0𝑑π‘₯.(3.17) Integrating the above inequality with respect to 𝑠 between 𝑑 and 𝑑+1, we get ξ€œΞ©||𝑀𝑛||(𝑑+1)π‘ž0ξ€œπ‘‘π‘₯≀𝐢𝑑𝑑+1ξ€œΞ©||𝑀𝑛||π‘ž0𝑑π‘₯π‘‘πœ‰.(3.18) Therefore, ξ€œΞ©||𝑀𝑛(||𝑑)π‘ž0𝑑π‘₯≀𝐢,βˆ€π‘‘β‰₯1.(3.19) Integrating (3.16) on [𝑑,𝑑+1] for 𝑑β‰₯1, we deduce that ξ€œπ‘‘π‘‘+1ξ€œΞ©ξ‚€||𝑀|βˆ‡π‘›||(π‘ž0βˆ’2)/2𝑀𝑛|2ξ€œπ‘‘π‘₯π‘‘πœ‰β‰€πΆπ‘‘π‘‘+1ξ€œΞ©||𝑀𝑛||(πœ‰)π‘ž0ξ€œπ‘‘π‘₯π‘‘πœ‰+𝐢Ω||𝑀𝑛||(𝑑)π‘ž0𝑑π‘₯≀𝐢.(3.20) Note that (3.20) insures that, for any 𝑑β‰₯1, there exists at least a 𝑑0∈[𝑑,𝑑+1] such that ξ€œΞ©|||βˆ‡ξ‚€||𝑀𝑛𝑑0ξ€Έ||(π‘ž0βˆ’2)/2𝑀𝑛𝑑0|||2𝑑π‘₯≀𝐢.(3.21) Standard Sobolev imbedding implies that ξ€œΞ©||𝑀𝑛𝑑0ξ€Έ||π‘ž0(𝑁/(π‘βˆ’2))𝑑π‘₯≀𝐢.(3.22) Now multiplying (3.5) by |𝑀𝑛|π‘ž1βˆ’2𝑀𝑛,π‘ž1=(𝑁/(π‘βˆ’2))π‘ž0, we have π‘βˆ’2π‘π‘ž0π‘‘ξ€œπ‘‘π‘‘Ξ©||𝑀𝑛||(𝑁/(π‘βˆ’2))π‘ž0𝑑π‘₯+𝐢𝑐,π‘ž0ξ€Έξ€œ,𝑁Ω|||βˆ‡ξ‚€||𝑀𝑛||(π‘π‘ž0βˆ’2𝑁+4)/(2π‘βˆ’4)𝑀𝑛|||2ξ€œπ‘‘π‘₯≀𝑙Ω||𝑀𝑛||(𝑁/(π‘βˆ’2))π‘ž0𝑑π‘₯.(3.23) Using HΓΆlder inequality, and Young inequality we deduce that ξ€œΞ©|𝑀𝑛|(𝑁/(π‘βˆ’2))π‘ž0𝑑π‘₯β‰€πœ€ξ…žξ€œΞ©|||βˆ‡ξ‚€||𝑀𝑛||(π‘π‘ž0βˆ’2𝑁+4)/(2π‘βˆ’4)𝑀𝑛|||2𝑑π‘₯+πΆπœ€ξ‚΅ξ€œΞ©||𝑀𝑛||π‘ž0𝑑π‘₯𝑁/(π‘βˆ’2).(3.24) Taking (3.24) into (3.23), it yields π‘‘ξ€œπ‘‘π‘‘Ξ©||𝑀𝑛||(𝑁/(π‘βˆ’2))π‘ž0𝑑x+πΆξ…žξ€·π‘,π‘ž0ξ€Έξ€œ,𝑁Ω|||βˆ‡ξ‚€||𝑀𝑛||(π‘π‘ž0βˆ’2𝑁+4)/(2π‘βˆ’4)𝑀𝑛|||2𝑑π‘₯β‰€πΆξ…žπœ€ξ‚΅ξ€œΞ©||𝑀𝑛||π‘ž0𝑑π‘₯𝑁/(π‘βˆ’2).(3.25) Integrating (3.25) between 𝑑0 and 𝑑0+𝑠,0<𝑠≀1, we have ξ€œΞ©||𝑀𝑛𝑑0ξ€Έ||+𝑠(𝑁/(π‘βˆ’2))π‘ž0ξ€œπ‘‘π‘₯≀Ω||𝑀𝑛𝑑0ξ€Έ||(𝑁/(π‘βˆ’2))π‘ž0𝑑π‘₯+πΆξ…žπœ€ξ‚΅ξ€œΞ©||𝑀𝑛||π‘ž0𝑑π‘₯𝑁/(π‘βˆ’2).(3.26) Therefore, from (3.19) and (3.22) we get ξ€œΞ©||𝑀𝑛(||𝑑)(𝑁/(π‘βˆ’2))π‘ž0ξ€œπ‘‘π‘₯=Ξ©||𝑀𝑛(||𝑑)π‘ž1𝑑π‘₯≀𝐢,βˆ€π‘‘β‰₯2,(3.27) with 𝐢 independent of 𝑛. Integrating (3.25) between 𝑑 and 𝑑+1 for 𝑑β‰₯2, we obtain ξ€œπ‘‘π‘‘+1ξ€œΞ©|||βˆ‡ξ‚€||𝑀𝑛||(π‘ž1βˆ’2)/2𝑀𝑛|||2ξ€œπ‘‘π‘₯π‘‘πœ‰=𝑑𝑑+1ξ€œΞ©||𝑀|βˆ‡(𝑛||(π‘π‘ž0βˆ’2𝑁+4)/(2π‘βˆ’4)𝑀𝑛)|2𝑑π‘₯π‘‘πœ‰β‰€πΆ.(3.28) Similar to (3.22), for any 𝑑β‰₯2, there exists at least a 𝑑0∈[𝑑,𝑑+1] such that ξ€œΞ©|𝑀𝑛(𝑑0)|π‘ž0(𝑁2/(π‘βˆ’2)2)𝑑π‘₯≀𝐢.(3.29) Bootstrap the above processes, we can deduce that ξ€œΞ©||𝑀𝑛(||𝑑)π‘žπ‘˜π‘‘π‘₯≀𝐢,βˆ€π‘‘β‰₯π‘‡π‘˜,(3.30) with π‘žπ‘˜=(𝑁/(π‘βˆ’2))π‘˜π‘ž0, and 𝐢 independent of 𝑛. Note that 𝑀𝑛→𝑀 in 𝐢([0,𝑇];𝐿1(Ξ©)) and π‘€βˆˆπΆ([0,𝑇];𝐿1(Ξ©)). From Lemma 2.6, we have ‖𝑀(𝑑)β€–π‘žπ‘˜πΏπ‘žπ‘˜(Ξ©)=ξ€œΞ©||||𝑀(𝑑)π‘žπ‘˜π‘‘π‘₯≀𝐢,βˆ€π‘‘β‰₯π‘‡π‘˜.(3.31) Taking π‘˜ large enough, we get the second part of (i) proved. If the integration are taken over [𝑑,𝑑+𝛿0] instead of [𝑑,𝑑+1], we get the first part of (i).
Now we are in the position to prove (ii). We multiply (3.5) with 𝑀𝑛 and deduce that 12π‘‘ξ€œπ‘‘π‘‘Ξ©||𝑀𝑛||2ξ€œπ‘‘π‘₯+𝑐Ω||βˆ‡π‘€π‘›||2ξ€œπ‘‘π‘₯≀𝑙Ω||𝑀𝑛||2𝑑π‘₯,(3.32) integrating over [𝑑,𝑑+1],𝑑β‰₯𝑇′, we get ξ€œπ‘‘π‘‘+1ξ€œΞ©||βˆ‡π‘€π‘›||2𝑑π‘₯𝑑𝑑≀𝐢,(3.33) with 𝐢 independent of 𝑛. Now, multiplying (3.5) with 𝑀𝑛𝑑, we obtain ξ€œΞ©||𝑀𝑛𝑑||2𝑑𝑑π‘₯+ξ€œπ‘‘π‘‘Ξ©||π‘Ž(π‘₯)βˆ‡π‘€π‘›||2𝑑𝑑π‘₯+ξ€œπ‘‘π‘‘Ξ©(𝐹(𝑀𝑛+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛)𝑑π‘₯=0,(3.34) where ∫𝐹(𝑣+𝜎)=𝜎0𝑓(𝑣+𝑠)𝑑𝑠. Integrating (3.34) between 𝑠 and 𝑑+1(𝑑≀𝑠<𝑑+1) gives ξ€œΞ©||βˆ‡π‘€π‘›(||𝑑+1)2ξ€œπ‘‘π‘₯+Ξ©(𝐹(𝑀𝑛(𝑑+1)+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛(ξ€œπ‘‘+1))𝑑π‘₯≀𝐢Ω||βˆ‡π‘€π‘›||(𝑠)2ξ€œπ‘‘π‘₯+𝐢Ω(𝐹(𝑀𝑛(𝑠)+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛(𝑠))𝑑π‘₯.(3.35) Now, integrating the above inequality with respect to 𝑠 between 𝑑 and 𝑑+1 we have ξ€œΞ©||βˆ‡π‘€π‘›(||𝑑+1)2ξ€œπ‘‘π‘₯+Ξ©(𝐹(𝑀𝑛(𝑑+1)+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛(ξ€œπ‘‘+1))𝑑π‘₯≀𝐢𝑑𝑑+1ξ€œΞ©||βˆ‡π‘€π‘›||2ξ€œπ‘‘π‘₯π‘‘πœ‰+𝐢𝑑𝑑+1||||ξ€œΞ©(𝐹(𝑀𝑛+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛||||)𝑑π‘₯π‘‘πœ‰.(3.36) Since ξ€œΞ©(𝐹(𝑀𝑛+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)π‘€π‘›ξ€œ)𝑑π‘₯=Ξ©ξ€œ10(𝑓(𝑣𝑛+𝑠𝑀𝑛)βˆ’π‘“(𝑣𝑛))𝑀𝑛𝑓𝑑𝑠𝑑π‘₯,(𝑣𝑛+πœπ‘€π‘›)βˆ’π‘“(π‘£π‘›ξ€œ)=10π‘“ξ…ž(𝑣𝑛+πœƒπœπ‘€π‘›)πœπ‘€π‘›π‘‘πœƒ,0β‰€πœβ‰€1.(3.37) We deduce that ||||ξ€œΞ©(𝐹(𝑀𝑛+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛||||β‰€ξ€œ)𝑑π‘₯Ξ©||π‘“ξ…žξ€·π‘£π‘›+πœξ…žπ‘€π‘›ξ€Έ||||𝑀𝑛||2𝑑π‘₯,0β‰€πœξ…žβ‰€1.(3.38) From the assumption (1.4) on 𝑓, we have ||||ξ€œΞ©(𝐹(𝑀𝑛+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛||||β‰€ξ€œ)𝑑π‘₯Ω𝐢||𝑣𝑛||π‘βˆ’2+||𝑀𝑛||π‘βˆ’2||𝑀+1𝑛||2𝑑π‘₯β‰€πΆβ€–π‘£π‘›β€–πΏπ‘βˆ’2π‘βˆ’1(Ξ©)+‖𝑀𝑛‖2𝐿2π‘βˆ’2(Ξ©)+‖𝑀𝑛‖𝑝𝐿𝑝(Ξ©).+1(3.39) Using results in (3.13) and (3.30), we know that ||||ξ€œΞ©(𝐹(𝑀𝑛+𝑣𝑛)βˆ’πΉ(𝑣𝑛)βˆ’π‘“(𝑣𝑛)𝑀𝑛||||)𝑑π‘₯≀𝐢,βˆ€π‘‘β‰₯π‘‡ξ…žξ…ž.(3.40) Set 𝑇0=max{π‘‡ξ…ž,π‘‡ξ…žξ…ž}. Combining (3.33), (3.34), (3.36), and (3.40) we have ξ€œΞ©||βˆ‡π‘€π‘›(||𝑑)2𝑑π‘₯≀𝐢,βˆ€π‘‘β‰₯𝑇0ξ€œ+1,(3.41)𝑑𝑑+1ξ€œΞ©||𝑀𝑛𝑑||2𝑑π‘₯π‘‘πœ‰β‰€πΆ,βˆ€π‘‘β‰₯𝑇0+1.(3.42) From Lemma 2.6, we obtain ‖𝑀‖𝐻10(Ξ©)≀𝐢,βˆ€π‘‘β‰₯𝑇0+1.(3.43) Thus we get the second part of (ii) proved. Taking integration over [𝑑,𝑑+𝛿0] instead of [𝑑,𝑑+1], the first part of (ii) follows. The proof is completed now.

4. Proof of the Main Theorems

Let {𝑆(𝑑)}𝑑β‰₯0 be the semigroup generated by problem (1.1) and let 𝑣(π‘₯) be an approximated solution to problem (2.3). Define 𝑆1𝑒(𝑑)0ξ€Έβˆ’π‘£(π‘₯)=𝑆(𝑑)𝑒0βˆ’π‘£(π‘₯).(4.1) Then it is easy to verify that {𝑆1(𝑑)}𝑑β‰₯0 is a continuous semigroup in (𝐿1(Ξ©)βˆ’π‘£) and hence in 𝐿1(Ξ©). From the results in Section 3, we know that the semigroup {𝑆(𝑑)}𝑑β‰₯0 possesses a global attractor π’œ in 𝐿1(Ξ©). 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(𝑑)}𝑑β‰₯0 possesses a global attractor π’œπ‘£ in πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©),1β‰€π‘Ÿ<∞, that is, π’œπ‘£ is compact, invariant in πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©),1β‰€π‘Ÿ<∞, and attracts every bounded initial set of 𝐿1(Ξ©) in the norm topology of πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©),1β‰€π‘Ÿ<∞.

Proof. From Theorem 3.1, we know that {𝑆(𝑑)}𝑑β‰₯0 possesses an absorbing set 𝐡0 in 𝐿1(Ξ©). Also {𝑆1(𝑑)}𝑑β‰₯0 possesses absorbing sets 𝐡1(=𝐡0βˆ’π‘£), 𝐡2, respectively, in 𝐿1(Ξ©) and πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©) for any 1β‰€π‘Ÿ<∞.
In the next, we prove the asymptotic compactness of {𝑆1(𝑑)}𝑑β‰₯0. Before that we establish the following estimate ξ€œΞ©||𝑀𝑛𝑑||2𝑑π‘₯≀𝐢,for𝑑largeenough.(4.2) Actually, differentiating (3.5) in time and denoting 𝑀𝑛=𝑀𝑛𝑑, we have ξ‚‹π‘€π‘›π‘‘ξ€·ξ‚‹π‘€βˆ’divπ‘Ž(π‘₯)βˆ‡π‘›ξ€Έ+𝑓′(𝑣𝑛+𝑀𝑛)𝑀𝑛=0.(4.3) Multiplying (4.3) by 𝑀𝑛 and using (1.3), we deduce that π‘‘ξ€œπ‘‘π‘‘Ξ©||𝑀𝑛||2ξ€œπ‘‘π‘₯≀𝑙Ω||𝑀𝑛||2𝑑π‘₯.(4.4) Integrating the above inequality between 𝑠 and 𝑑+1(𝑑≀𝑠<𝑑+1) gives ξ€œΞ©||𝑀𝑛||(𝑑+1)2ξ€œπ‘‘π‘₯≀2𝑙𝑠𝑑+1ξ€œΞ©||𝑀𝑛||2ξ€œπ‘‘π‘₯π‘‘πœ‰+Ξ©||𝑀𝑛||(𝑠)2𝑑π‘₯.(4.5) Integrating the above inequality with respect to 𝑠 between 𝑑 and 𝑑+1, using (3.42) we obtain ξ€œΞ©||𝑀𝑛||(𝑑)2ξ€œπ‘‘π‘₯≀(2𝑙+1)𝑑𝑑+1ξ€œΞ©||𝑀𝑛||2𝑑π‘₯π‘‘πœ‰β‰€(2𝑙+1)𝐢,βˆ€π‘‘β‰₯𝑇0+1.(4.6) Now we prove the asymptotic compactness of the semigroup {𝑆1(𝑑)}𝑑β‰₯0, that is, for any sequences {𝑀0,π‘˜}βŠ‚π΅1,π‘‘π‘˜β†’βˆž, sequence {π‘€π‘˜(π‘‘π‘˜)} has convergent subsequences, where π‘€π‘˜(𝑑)=𝑆1(𝑑)𝑀0,π‘˜. Since {𝑆1(𝑑)}𝑑β‰₯0 is compact in πΏπ‘Ÿ(Ξ©),1β‰€π‘Ÿ<∞, there is a subsequence {π‘€π‘˜π‘–(π‘‘π‘˜π‘–)}, which is a Cauchy sequence in πΏπ‘Ÿ(Ξ©),1β‰€π‘Ÿ<∞. Denoting π‘’π‘˜π‘–(π‘‘π‘˜π‘–)=π‘€π‘˜π‘–(π‘‘π‘˜π‘–)+𝑣,π‘’π‘˜π‘—(π‘‘π‘˜π‘—)=π‘€π‘˜π‘—(π‘‘π‘˜π‘—)+𝑣, we deduce that π‘β€–β€–π‘€π‘˜π‘–(π‘‘π‘˜π‘–)βˆ’π‘€π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚β€–β€–2𝐻10(Ξ©)≀𝑀divπ‘Ž(π‘₯)βˆ‡π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ€Έξ‚€ξ‚€π‘€+π‘£ξ€Έξ€Έβˆ’divπ‘Ž(π‘₯)βˆ‡π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚+𝑣,π‘€π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ€Έβˆ’π‘€π‘˜π‘—ξ‚€π‘‘π‘˜π‘—=ξ‚¬πœ•ξ‚ξ‚­π‘‘π‘€π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ€Έβˆ’πœ•π‘‘π‘€π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚ξ€·π‘’+π‘“π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ‚€π‘’ξ€Έξ€Έβˆ’π‘“π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚ξ‚,π‘€π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ€Έβˆ’π‘€π‘˜π‘—ξ‚€π‘‘π‘˜π‘—β‰€β€–β€–πœ•ξ‚ξ‚­π‘‘π‘€π‘˜π‘–(π‘‘π‘˜π‘–)βˆ’πœ•π‘‘π‘€π‘˜π‘—(π‘‘π‘˜π‘—)‖‖𝐿2(Ξ©)β€–β€–π‘€π‘˜π‘–(π‘‘π‘˜π‘–)βˆ’π‘€π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚β€–β€–πΏ2(Ξ©)+β€–β€–π‘“ξ€·π‘€π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ€Έξ€Έξ‚€π‘€+π‘£βˆ’π‘“π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚ξ‚β€–β€–+π‘£πΏπœŽ(Ξ©)β€–β€–π‘€π‘˜π‘–ξ€·π‘‘π‘˜π‘–ξ€Έβˆ’π‘€π‘˜π‘—ξ‚€π‘‘π‘˜π‘—ξ‚β€–β€–πΏπœŽβ€²(Ξ©).(4.7) We then conclude form (4.6) that {π‘€π‘˜π‘–(π‘‘π‘˜π‘–)} is a Cauchy sequence in 𝐻10(Ξ©), and thus {𝑆1(𝑑)}𝑑β‰₯0 is asymptotically compact in πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©),1β‰€π‘Ÿ<∞.
Using Lemma 2.7, we conclude that {𝑆1(𝑑)}𝑑β‰₯0 possesses a global attractor π’œπ‘£, which is compact, invariant in πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©), and attracts every bounded initial sets of 𝐿1(Ξ©) in the topology of πΏπ‘Ÿ(Ξ©)∩𝐻10(Ξ©).

Completion of the Proof of Theorem 1.1
Note that π’œπ‘£=βˆ©π‘ β‰₯0βˆͺ𝑑β‰₯𝑠𝑆1(𝑑)𝐡2𝐿1(Ξ©)=βˆ©π‘ β‰₯0βˆͺ𝑑β‰₯𝑠𝑆1(𝑑)𝐡1𝐿1(Ξ©)=βˆ©π‘ β‰₯0βˆͺ𝑑β‰₯𝑠𝑆(𝑑)𝐡0ξ€Έβˆ’π‘£πΏ1(Ξ©),π’œ=βˆ©π‘ β‰₯0βˆͺ𝑑β‰₯𝑠𝑆(𝑑)𝐡0𝐿1(Ξ©).(4.8) Thus we have π’œ=π’œπ‘£+𝑣.(4.9) The above relation between π’œ and π’œπ‘£ implies the conclusion of Theorem 1.1 directly.

Proof of Theorem 1.2. Let 𝑒(𝑑),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 𝑒(𝑑) converges to 𝑣 in 𝐿1(Ξ©) as π‘‘β†’βˆž. Using arguments similar to Section 3, we can prove similar regularity results for 𝑀(=π‘’βˆ’π‘£) and then prove the asymptotic compactness of the semigroup 𝑆1(𝑑) as in Theorem 4.1. Thus, we obtain that 𝑀(𝑑) converges to 0 in πΏπ‘Ÿ(Ξ©),1β‰€π‘Ÿ<∞, as π‘‘β†’βˆž. Moreover from the asymptotic compactness of the semigroup 𝑆1(𝑑), we know that 𝑀(𝑑) converges to 0 in 𝐻10(Ξ©) as π‘‘β†’βˆž. Else, we have a sequence π‘‘π‘›β†’βˆž, such that 𝐢>‖𝑀(𝑑𝑛)‖𝐻10(Ξ©)β‰₯πœ–>0. Since the semigroup 𝑆1(𝑑) is asymptotically compact, there is a subsequence π‘‘π‘›π‘—β†’βˆž, such that 𝑀(𝑑𝑛𝑗) converges to a function πœ’ in 𝐻10(Ξ©) and hence in 𝐿1(Ξ©). Thus πœ’=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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