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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 272145, 20 pages
http://dx.doi.org/10.1155/2012/272145
Research Article

Regularity and Exponential Growth of Pullback Attractors for Semilinear Parabolic Equations Involving the Grushin Operator

Faculty of Applied Mathematics and Informatics, Hanoi University of Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam

Received 24 November 2011; Accepted 28 December 2011

Academic Editor: ShaoyongΒ Lai

Copyright Β© 2012 Nguyen Dinh Binh. 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

Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain Ξ©. We prove the regularity and exponential growth of a pullback attractor in the space 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©) for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.

1. Introduction

Let Ξ© be a bounded domain in ℝ𝑁1×ℝ𝑁2(𝑁1,𝑁2β‰₯1), with smooth boundary πœ•Ξ©. In this paper, we consider the following problem: π‘’π‘‘βˆ’πΊπ‘ π‘’+𝑓(𝑒)=𝑔(𝑑,π‘₯),(𝑑,π‘₯)βˆˆπ‘„πœ,𝑇]],=(𝜏,𝑇×Ω,𝑒(π‘₯,𝑑)=0,π‘₯βˆˆπœ•Ξ©,π‘‘βˆˆ(𝜏,𝑇𝑒(π‘₯,𝜏)=π‘’πœ(π‘₯),π‘₯∈Ω,(1.1) where 𝐺𝑠𝑒=Ξ”π‘₯1||π‘₯𝑒+1||2𝑠Δπ‘₯2ξ€·π‘₯𝑒,π‘₯=1,π‘₯2ξ€ΈβˆˆΞ©βŠ‚β„π‘1×ℝ𝑁2,𝑠⩾0,(1.2) is the Grushin operator, π‘’πœβˆˆπΏ2(Ξ©) is given, the nonlinearity 𝑓 and the external force 𝑔 satisfy the following conditions.(H1) The nonlinearity π‘“βˆˆπΆ1(ℝ,ℝ) satisfies𝑓(𝑒)𝑒β‰₯𝐢1|𝑒|π‘βˆ’πΆ2||||,𝑝β‰₯2,(1.3)𝑓′(𝑒)≀𝐢3|𝑒|π‘βˆ’2+𝐢4,(1.4)𝑓′(𝑒)β‰₯βˆ’β„“,(1.5) where β„“, 𝐢𝑖, (𝑖=1,2,3,4) are positive constants. Relation (1.3) and (1.4) imply that𝛼1||𝑒|π‘βˆ’π›Ό2≀𝐹(𝑒)≀𝛼3||𝑒|𝑝+𝛼4,(1.6) where ∫𝐹(𝑠)=𝑠0𝑓(𝜏)π‘‘πœ, and 𝛼𝑖(𝑖=1,2,3,4) are positive constants.(H2)π‘”βˆˆπ‘Š1,2loc(ℝ;𝐿2(Ξ©)) satisfiesξ€œπ‘‘βˆ’βˆžπ‘’πœ†1𝑠‖𝑔(𝑠)β€–2𝐿2(Ξ©)+‖𝑔′(𝑠)β€–2𝐿2(Ξ©)𝑑𝑠<+∞,βˆ€π‘‘βˆˆβ„,(1.7) where πœ†1 is the first eigenvalue of the operator βˆ’πΊπ‘  in Ξ© with the homogeneous Dirichlet boundary condition.

The Grushin operator 𝐺𝑠 was first introduced in [1]. Noting that if 𝑠>0, then 𝐺𝑠 is not elliptic in domains of ℝ𝑁1×ℝ𝑁2 which intersect the hyperplane {π‘₯1=0}. In the last few years, the existence and long-time behavior of solutions to parabolic equations involving the Grushin operator have been studied widely in both autonomous and nonautonomous cases (see, e.g., [2–7]). In particular, the existence of a pullback attractor in 𝑆10(Ξ©)βˆ©πΏπ‘(Ξ©) for the process associated to problem (1.1) is considered in [2].

In this paper we continue the study in the paper [2]. First, we will prove the existence of pullback attractors in 𝑆20(Ξ©) (see Section 2 for its definition) and 𝐿2π‘βˆ’2(Ξ©). As we know, if the external force 𝑔 is only in 𝐿2(Ξ©), then solutions of problem (1.1) are at most in 𝐿2π‘βˆ’2(Ξ©)βˆ©π‘†20(Ξ©) and have no higher regularity. Therefore, there are no compact embedding results that hold for this case. To overcome the difficulty caused by the lack of embedding results, we exploit the asymptotic a priori estimate method which was initiated in [8, 9] for autonomous equations and developed recently for nonautonomous equations in the case of pullback attractors in [10]. Noting that, to prove the existence of pullback attractors in 𝑆10(Ξ©)βˆ©πΏπ‘(Ξ©), we only need assumption (H2) of the external force 𝑔; however, to prove the existence of pullback attractors in 𝑆20(Ξ©) and 𝐿2π‘βˆ’2(Ξ©), we need an additional assumption of 𝑔, namely, (3.18) in Section 3. Next, following the general lines of the approach in [11], we give exponential growth conditions in 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©) for the pullback attractors. It is noticed that, as far as we know, the best known results on the pullback attractors for nonautonomous reaction-diffusion equations are the boundedness and exponential growth in 𝐻2(Ξ©) of the pullback attractors [11, 12]. Therefore, the obtained results seem to be optimal and, in particular when 𝑠=0, improve the recent results on pullback attractors for the nonautonomous reaction-diffusion equations in [11–15].

The content of the paper is as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on function spaces and pullback attractors which we will use. In Section 3, we prove the existence of pullback attractors in the spaces 𝑆20(Ξ©) and 𝐿2π‘βˆ’2(Ξ©) by using the asymptotic a priori estimate method. In Section 4, under additional assumptions of 𝑔, an exponential growth in 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©) for the pullback attractors is deduced.

2. Preliminaries

2.1. Operator and Function Spaces

In order to study the boundary value problem for equations involving the Grushin operator, we have usually used the natural energy space 𝑆10(Ξ©) defined as the completion of 𝐢∞0(Ξ©) in the following norm: ‖𝑒‖𝑆10(Ξ©)=ξ‚΅ξ€œΞ©ξ‚΅|||βˆ‡π‘₯1𝑒|||2+||π‘₯1||2𝑠||βˆ‡π‘₯2𝑒||2𝑑π‘₯1/2(2.1)

and the scalar product ξ‚΅ξ€œ((𝑒,𝑣))∢=Ξ©ξ‚€βˆ‡π‘₯1π‘’βˆ‡π‘₯1||π‘₯𝑣+1||2π‘ βˆ‡π‘₯2π‘’βˆ‡π‘₯2𝑣𝑑π‘₯1/2.(2.2)The following lemma comes from [16].

Lemma 2.1. Assume that Ξ© is a bounded domain in ℝ𝑁1×ℝ𝑁2(𝑁1,𝑁2β‰₯0). Then the following embeddings hold:(i)𝑆10(Ξ©)β†ͺ𝐿2βˆ—π›Ό(Ξ©) continuously;(ii)𝑆10(Ξ©)β†ͺ𝐿𝑝(Ξ©) compactly if π‘βˆˆ[1,2βˆ—π‘ ), where2βˆ—π‘ =2𝑁(𝑠)/(𝑁(𝑠)βˆ’2), 𝑁(𝑠)=𝑁1+(𝑠+1)𝑁2.

Now, we introduce the space 𝑆20(Ξ©) defined as the closure of 𝐢∞0(Ξ©) with the norm ‖𝑒‖𝑆20(Ξ©)=ξ‚΅ξ€œΞ©ξ‚€||Ξ”π‘₯1𝑒||2+||π‘₯1||2s||Ξ”π‘₯2𝑒||2𝑑π‘₯1/2=ξ‚΅ξ€œΞ©||Gs𝑒||2𝑑π‘₯1/2.(2.3)

The following lemma comes directly from the definitions of 𝑆10(Ξ©) and 𝑆20(Ξ©).

Lemma 2.2. Assume that Ξ© is a bounded domain in ℝ𝑁1×ℝ𝑁2(𝑁1,𝑁2β‰₯0), with smooth boundary πœ•Ξ©. Then 𝑆20(Ξ©)βŠ‚π‘†10(Ξ©) continuously.

It is known that (see, e.g., [3]) for the operator 𝐴=βˆ’πΊπ‘ , there exist {𝑒𝑗}𝑗β‰₯1

such that𝑒𝑗,π‘’π‘˜ξ€Έ=π›Ώπ‘—π‘˜,𝐴𝑒𝑗=πœ†π‘—π‘’π‘—,𝑗,π‘˜=1,2,…,0<πœ†1β‰€πœ†2β‰€πœ†3≀⋯,πœ†π‘—βŸΆ+∞asπ‘—βŸΆβˆž,(2.4)

and {𝑒𝑗}𝑗β‰₯1 is a complete orthonormal system in 𝐿2(Ξ©).

2.2. Pullback Attractors

Let 𝑋 be a Banach space with the norm β€–β‹…β€–. ℬ(𝑋) denotes all bounded sets of 𝑋. The Hausdorff semidistance between 𝐴 and 𝐡 is defined bydist(𝐴,𝐡)=supπ‘₯∈𝐴infπ‘¦βˆˆπ΅β€–π‘₯βˆ’π‘¦β€–.(2.5)

Let {π‘ˆ(𝑑,𝜏)βˆΆπ‘‘β‰₯𝜏,πœβˆˆβ„} be a process in 𝑋, that is, π‘ˆ(𝑑,𝜏)βˆΆπ‘‹β†’π‘‹ such that π‘ˆ(𝜏,𝜏)=𝐼𝑑 and π‘ˆ(𝑑,𝑠)π‘ˆ(𝑠,𝜏)=π‘ˆ(𝑑,𝜏) for all 𝑑β‰₯𝑠β‰₯𝜏,β€‰πœβˆˆβ„. The process {π‘ˆ(𝑑,𝜏)} is said to be norm-to-weak continuous if π‘ˆ(𝑑,𝜏)π‘₯π‘›β‡€π‘ˆ(𝑑,𝜏)π‘₯, as π‘₯𝑛→π‘₯ in 𝑋, for all 𝑑β‰₯𝜏,β€‰πœβˆˆβ„. The following result is useful for proving the norm-to-weak continuity of a process.

Proposition 2.3 (see [9]). Let 𝑋,π‘Œ be two Banach spaces, and let π‘‹βˆ—,π‘Œβˆ— be, respectively, their dual spaces. Suppose that 𝑋 is dense in π‘Œ, the injection π‘–βˆΆπ‘‹β†’π‘Œ is continuous, and its adjoint π‘–βˆ—βˆΆπ‘Œβˆ—β†’π‘‹βˆ— is dense, and {π‘ˆ(𝑑,𝜏)} is a continuous or weak continuous process on π‘Œ. Then {π‘ˆ(𝑑,𝜏)} is norm-to-weak continuous on 𝑋 if and only if for 𝑑β‰₯𝜏, πœβˆˆβ„, π‘ˆ(𝑑,𝜏) maps compact sets of 𝑋 into bounded sets of 𝑋.

Definition 2.4. The process {π‘ˆ(𝑑,𝜏)} is said to be pullback asymptotically compact if for any π‘‘βˆˆβ„, any π·βˆˆβ„¬(𝑋), any sequence πœπ‘›β†’βˆ’βˆž, and any sequence {π‘₯𝑛}βŠ‚π·, the sequence {π‘ˆ(𝑑,πœπ‘›)π‘₯𝑛} is relatively compact in 𝑋.

Definition 2.5. A family of bounded sets ℬ={𝐡(𝑑)βˆΆπ‘‘βˆˆβ„}βŠ‚π‘‹ is called a pullback absorbing set for the process {π‘ˆ(𝑑,𝜏)} if for any π‘‘βˆˆβ„ and any π·βˆˆβ„¬(𝑋), there exist 𝜏0=𝜏0(𝐷,𝑑)≀𝑑 and 𝐡(𝑑)βˆˆβ„¬ such that ξšπœβ‰€πœ0π‘ˆ(𝑑,𝜏)π·βŠ‚π΅(𝑑).(2.6)

Definition 2.6. The family π’œ={𝐴(𝑑)βˆΆπ‘‘βˆˆβ„}βŠ‚β„¬(𝑋) is said to be a pullback attractor for {π‘ˆ(𝑑,𝜏)} if (1)𝐴(𝑑) is compact for all π‘‘βˆˆβ„, (2)π’œ is invariant, that is, π‘ˆ(𝑑,𝜏)𝐴(𝜏)=𝐴(𝑑),βˆ€π‘‘β‰₯𝜏,(2.7)(3)π’œ is pullback attracting, that is, limπœβ†’βˆ’βˆždist(π‘ˆ(𝑑,𝜏)𝐷,𝐴(𝑑))=0,βˆ€π·βˆˆβ„¬(𝑋),andallπ‘‘βˆˆβ„,(2.8)(4)if {𝐢(𝑑)βˆΆπ‘‘βˆˆβ„} is another family of closed pullback attracting sets, then 𝐴(𝑑)βŠ‚πΆ(𝑑), for all π‘‘βˆˆβ„.

Theorem 2.7 (see [13]). Let {π‘ˆ(𝑑,𝜏)} be a norm-to-weak continuous process which is pullback asymptotically compact. If there exists a pullback absorbing set ℬ={𝐡(𝑑)βˆΆπ‘‘βˆˆβ„}, then {π‘ˆ(𝑑,𝜏)} has a unique pullback attractor π’œ={𝐴(𝑑)βˆΆπ‘‘βˆˆβ„} and 𝐴(𝑑)=π‘ β‰€π‘‘ξšπœβ‰€π‘ π‘ˆ(𝑑,𝜏)𝐡(𝜏).(2.9)

In the rest of the paper, we denote by |β‹…|2, (β‹…,β‹…) the norm and inner product in 𝐿2(Ξ©), respectively, and by |β‹…|𝑝 the norm in 𝐿𝑝(Ξ©). By β€–β‹…β€– we denote the norm in 𝑆10(Ξ©). For a Banach space 𝐸, ‖⋅‖𝐸 will be the norm. We also denote by 𝐢 an arbitrary constant, which is different from line to line, and even in the same line.

3. Existence of Pullback Attractors in 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©)

It is well known (see, e.g., [2] or [14]) that under conditions (𝐻1)βˆ’(𝐻2), problem (1.1) defines a processπ‘ˆ(𝑑,𝜏)∢𝐿2(Ξ©)βŸΆπ‘†10(Ξ©)βˆ©πΏπ‘(Ξ©),βˆ€π‘‘β‰₯𝜏,(3.1)where π‘ˆ(𝑑,𝜏)π‘’πœ is the unique weak solution of (1.1) with initial datum π‘’πœ at time 𝜏. The process {π‘ˆ(𝑑,𝜏)} has a pullback attractor in 𝑆10(Ξ©)βˆ©πΏπ‘(Ξ©).

In this section, we will prove that the pullback attractor is in fact in 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©).

Lemma 3.1. Assuming that 𝑓 and 𝑔 satisfy (H 1)-(H 2), 𝑒(𝑑) is a weak solution of (1.1). Then the following inequality holds for 𝑑>𝜏: ‖𝑒‖2+|𝑒|π‘π‘ξ‚΅π‘’β‰€πΆβˆ’πœ†1(π‘‘βˆ’πœ)||π‘’πœ||22+1+π‘’βˆ’πœ†1π‘‘ξ€œπ‘‘βˆ’βˆžπ‘’πœ†1𝑠||||𝑔(𝑠)22ξ‚Ά,𝑑𝑠(3.2) where 𝐢 is a positive constant.

Proof. Multiplying (1.1) by 𝑒 and then integrating over Ξ©, we get 12𝑑𝑑𝑑|𝑒|22+‖𝑒‖2+ξ€œΞ©ξ€œπ‘“(𝑒)𝑒𝑑π‘₯=Ξ©1𝑔(𝑑)𝑒𝑑π‘₯β‰€πœ†1||||𝑔(𝑑)22+πœ†14|𝑒|22.(3.3) Using hypothesis (H1) and the inequality ‖𝑒‖2β‰₯πœ†1|𝑒|22, we have 𝑑𝑑𝑑|𝑒|22+πœ†1|𝑒|22ξ€·+𝐢‖𝑒‖2+|𝑒|𝑝𝑝||||≀𝐢1+𝑔(𝑑)22.(3.4) Letting ∫𝐹(𝑠)=𝑠0𝑓(𝜏)π‘‘πœ, by (H1), we have 𝛼1|𝑒|π‘βˆ’π›Ό2≀𝐹(𝑒)≀𝛼3|𝑒|𝑝+𝛼4.(3.5) Now multiplying (3.4) by π‘’πœ†1𝑑 and using (3.5), we get π‘‘ξ‚€π‘’π‘‘π‘‘πœ†1𝑑||||𝑒(𝑑)22+πΆπ‘’πœ†1𝑑‖𝑒‖2ξ€œ+2Ω𝐹(𝑒(𝑑))𝑑π‘₯β‰€πΆπ‘’πœ†1𝑑||||1+𝑔(𝑑)22.(3.6) Integrating (3.6) from 𝜏 to π‘ βˆˆ[𝜏,π‘‘βˆ’1] and 𝑠 to 𝑠+1, respectively, we obtain π‘’πœ†1𝑠||||𝑒(𝑠)22β‰€π‘’πœ†1𝜏||π‘’πœ||22+πΆπ‘’πœ†1π‘ ξ€œ+πΆπ‘ πœπ‘’πœ†1π‘Ÿ||||𝑔(π‘Ÿ)22[],πΆξ€œπ‘‘π‘Ÿ,βˆ€π‘ βˆˆπœ,π‘‘βˆ’1(3.7)𝑠𝑠+1π‘’πœ†1π‘Ÿξ‚΅β€–π‘’(π‘Ÿ)β€–2ξ€œ+2Ω𝐹(𝑒(π‘₯,π‘Ÿ))𝑑π‘₯π‘‘π‘Ÿβ‰€π‘’πœ†1𝑠||||𝑒(𝑠)22ξ€œ+𝐢𝑠𝑠+1π‘’πœ†1π‘Ÿξ‚€||||1+𝑔(π‘Ÿ)22ξ‚π‘‘π‘Ÿβ‰€π‘’πœ†1𝜏||π‘’πœ||22+πΆπ‘’πœ†1π‘ ξ€œ+πΆπ‘ πœπ‘’πœ†1π‘Ÿ||||𝑔(π‘Ÿ)22π‘‘π‘Ÿ+πΆπ‘’πœ†1(𝑠+1)ξ€œ+C𝑠𝑠+1π‘’πœ†1π‘Ÿ||||𝑔(π‘Ÿ)22ξ‚΅π‘’π‘‘π‘Ÿβ‰€πΆπœ†1𝜏||π‘’πœ||22+π‘’πœ†1𝑑+ξ€œπ‘‘πœπ‘’πœ†1π‘Ÿ||||𝑔(π‘Ÿ)22ξ‚Ά.π‘‘π‘Ÿ(3.8) Multiplying (1.1) by 𝑒𝑑 and integrating over Ξ©, we have ||𝑒𝑑||(𝑠)22+12𝑑𝑑𝑠‖𝑒(𝑠)β€–2ξ€œ+2Ξ©ξ‚Ά=ξ€œπΉ(𝑒(π‘₯,𝑠))𝑑π‘₯Ω𝑔(𝑠)𝑒𝑑1(𝑠)≀2||||𝑔(𝑠)22+12||𝑒𝑑||(𝑠)22.(3.9) Thus π‘’πœ†1𝑠||𝑒𝑑||(𝑠)22+π‘‘ξ€Ίπ‘’π‘‘π‘ πœ†1𝑠‖‖𝑒(𝑠)2∫+2Ω𝐹(𝑒(π‘₯,𝑠))𝑑π‘₯ξ€Έξ€»β‰€πœ†1π‘’πœ†1𝑠‖𝑒(𝑠)β€–2ξ€œ+2Ω𝐹(𝑒(π‘₯,𝑠))𝑑π‘₯+π‘’πœ†1𝑠||||𝑔(𝑠)22.(3.10) Combining (3.8) and (3.10), and using the uniform Gronwall inequality, we have π‘’πœ†1𝑑(‖𝑒𝑑)β€–2ξ€œ+2Ω𝑒𝐹(𝑒(π‘₯,𝑑))𝑑π‘₯β‰€πΆπœ†1𝜏||π‘’πœ||2+π‘’πœ†1𝑑+ξ€œπ‘‘βˆ’βˆžπ‘’πœ†1𝑠||||𝑔(𝑠)22𝑑𝑠.(3.11) Using (H1) once again and thanks to π‘’πœ†1𝜏|π‘’πœ|22β†’0 as πœβ†’βˆ’βˆž, we get the desired result from (3.11).

Lemma 3.2. Assume that (H 1), (H 2) hold. Then for any π‘‘βˆˆβ„ and any π·βŠ‚πΏ2(Ξ©) that is bounded, there exists 𝜏0β‰€π‘‘βˆ’1 such that ||𝑒𝑑(||𝑑)22≀𝐢1+π‘’βˆ’πœ†1π‘‘ξ€œπ‘‘βˆ’βˆžπ‘’πœ†1𝑠||||𝑔(𝑠)22+||||𝑔′(𝑠)22,𝑑𝑠(3.12) for any πœβ‰€πœ0 and any π‘’πœβˆˆπ·, where 𝑒𝑑(𝑠)=(𝑑/𝑑𝑑)(π‘ˆ(𝑑,𝜏)π‘’πœ)|𝑑=𝑠.

Proof. Integrating (3.10) from π‘Ÿ to π‘Ÿ+1,π‘Ÿβˆˆ[𝜏,π‘‘βˆ’1] and using (3.8) and (3.11), in particular we find ξ€œπ‘Ÿπ‘Ÿ+1π‘’πœ†1s||𝑒𝑑||22π‘‘π‘ β‰€π‘’πœ†1π‘Ÿξ‚΅β€–β€–π‘’(π‘Ÿ)2ξ€œ+2Ω𝐹(𝑒(π‘₯,π‘Ÿ))𝑑π‘₯+πœ†1ξ€œπ‘Ÿπ‘Ÿ+1π‘’πœ†1𝑠‖𝑒(𝑠)β€–2ξ€œ+2Ξ©ξ‚Ά+ξ€œπΉ(𝑒(π‘₯,𝑠))𝑑π‘₯π‘‘π‘ π‘Ÿπ‘Ÿ+1π‘’πœ†1𝑠||||𝑔(𝑠)22ξ‚΅π‘’π‘‘π‘ β‰€πΆπœ†1𝜏||π‘’πœ||22+π‘’πœ†1𝑑+ξ€œπ‘‘βˆ’βˆžπ‘’πœ†1𝑠||||𝑔(𝑠)22ξ‚Ά.𝑑𝑠(3.13) On the other hand, differentiating (1.1) and denoting  𝑣=𝑒𝑑, we have π‘£π‘‘βˆ’πΊπ‘ π‘£+π‘“ξ…ž(𝑒)𝑣=π‘”ξ…ž(𝑑).(3.14) Taking the inner product of (3.14) with 𝑣 in 𝐿2(Ξ©), we get 12𝑑𝑑𝑑|𝑣|22+‖𝑣‖2𝑔+(𝑓′(𝑒)𝑣,𝑣)=ξ…žξ€Έ.(𝑑),𝑣(3.15) Using (1.5) and Young’s inequality, after a few computations, we see that π‘‘ξ€·π‘’π‘‘π‘‘πœ†1𝑑|𝑣|22ξ€Έ+2π‘’πœ†1𝑑‖𝑣‖2β‰€πΆπ‘’πœ†1𝑑|𝑣|22+πΆπ‘’πœ†1𝑑||π‘”ξ…ž||(𝑑)22.(3.16) Combining (3.16) and (3.13) and using the uniform Gronwall inequality, we obtain π‘’πœ†1𝑑||||𝑣(𝑑)22ξ‚΅π‘’β‰€πΆπœ†1𝜏||π‘’πœ||22+π‘’πœ†1𝑑+ξ€œπ‘‘βˆ’βˆžπ‘’πœ†1𝑠||||𝑔(𝑠)22+||π‘”ξ…ž||(𝑠)22.𝑑𝑠(3.17) The proof is now complete because π‘’πœ†1𝜏|π‘’πœ|22β†’0 as πœβ†’βˆ’βˆž.

3.1. Existence of a Pullback Attractor in𝐿2π‘βˆ’2(Ξ©)

In this section, following the general lines of the method introduced in [9], we prove the existence of a pullback attractor in 𝐿2π‘βˆ’2(Ξ©). In order to do this, we need an additional condition of π‘”ξ€œπ‘‘βˆ’βˆžπ‘’πœ†1π‘‘β€–β€–π‘”ξ…žβ€–β€–(𝑑)π‘šβ€²πΏπ‘š(Ξ©)𝑑𝑑<+∞,βˆ€tβˆˆβ„,(3.18) where π‘š, π‘šξ…ž are defined as in (3.30).

Lemma 3.3. The process {π‘ˆ(𝑑,𝜏)} associated to problem (1.1) has a pullback absorbing set in 𝐿2π‘βˆ’2(Ξ©).

Proof. Multiplying (1.1) by |𝑒|pβˆ’2𝑒 and integrating over Ξ©, we get ξ€œΞ©π‘’π‘‘|𝑒|π‘βˆ’2ξ€œπ‘’π‘‘π‘₯+(π‘βˆ’1)Ξ©ξ‚€||βˆ‡π‘₯1𝑒||2+||π‘₯1||2s||βˆ‡π‘₯2𝑒||2|𝑒|π‘βˆ’2ξ€œπ‘‘π‘₯+Ξ©|||𝑓(𝑒)𝑒|π‘βˆ’2||=ξ€œπ‘’π‘‘π‘₯Ξ©||||𝑔(𝑑)π‘’π‘βˆ’2𝑒𝑑π‘₯.(3.19) From (1.3) and the fact that 𝐿𝑝(Ξ©)βŠ‚πΏπ‘βˆ’2(Ξ©) continuously, we have ξ€œΞ©π‘“(𝑒)|𝑒|π‘βˆ’2ξ€œπ‘’π‘‘π‘₯β‰₯Ω𝐢1|𝑒|π‘βˆ’πΆ2ξ€Έ|𝑒|π‘βˆ’2𝑑π‘₯β‰₯𝐢1‖𝑒‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)βˆ’πΆ2|𝑒|𝑝𝑝.(3.20) On the other hand, by Cauchy’s inequality, we see that ||||ξ€œΞ©π‘’π‘‘|𝑒|π‘βˆ’2||||≀𝐢𝑒𝑑π‘₯14‖𝑒‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+1𝐢1||𝑒𝑑||22,(3.21)||||ξ€œΞ©π‘”(𝑑)|𝑒|π‘βˆ’2||||≀𝐢𝑒𝑑π‘₯14‖𝑒‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+14||𝑔||(𝑑)22.(3.22) Combining (3.19)–(3.22) imply that ‖𝑒‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)ξ‚€||𝑒≀𝐢𝑑||22+|𝑒|𝑝𝑝+||||𝑔(𝑑)22.(3.23) Applying (3.2) and Lemma 3.2, we conclude the existence of a pullback absorbing set in 𝐿2π‘βˆ’2(Ξ©) for the process π‘ˆ(𝑑,𝜏).

Lemma 3.4. For any π‘ βˆˆβ„, any 2≀𝑝<∞, and any bounded set π΅βŠ‚πΏ2(Ξ©), there exists 𝜏0 such that ξ€œΞ©||𝑒𝑑||(𝑠)𝑝𝑑π‘₯≀𝑀,βˆ€πœβ‰€πœ0,π‘’πœβˆˆπ΅,(3.24) where 𝑀 depends on 𝑠, 𝑝 but not on 𝐡, and 𝑒𝑑(𝑠)=(𝑑/𝑑𝑑)(π‘ˆ(𝑑,𝜏)π‘’πœ)|𝑑=𝑠.

Proof. We will prove the lemma by induction argument. Letting  𝛽=𝑁(𝑠)/(𝑁(𝑠)βˆ’2)>1 and denoting 𝑣=𝑒𝑑 we prove that for π‘˜=0,1,2,…, there exist πœπ‘˜ and π‘€π‘˜(𝑠) such that π‘’πœ†1π‘ ξ€œΞ©||||𝑣(𝑠)2π›½π‘˜π‘‘π‘₯β‰€π‘€π‘˜(s)foranyπ‘’πœβˆˆπ΅,πœβ‰€πœπ‘˜,(π‘ƒπ‘˜)ξ€œπ‘ π‘ +1ξ‚΅π‘’πœ†1π‘Ÿξ€œΞ©||||𝑣(π‘Ÿ)2π›½π‘˜+1𝑑π‘₯1/π›½π‘‘π‘Ÿβ‰€π‘€π‘˜(𝑠)foranyπ‘’πœβˆˆπ΅,πœβ‰€πœπ‘˜,(π‘„π‘˜) where πœπ‘˜ depends on π‘˜ and 𝐡 and π‘€π‘˜ depends only on π‘˜.
For π‘˜=0, we have (𝑃0) from (3.17). Integrating (3.16) and using 𝑆10(Ξ©)β†ͺ𝐿2𝛽(Ξ©) continuously, we get (𝑄0).
Assuming that (π‘ƒπ‘˜), (π‘„π‘˜) hold, we prove so are (π‘ƒπ‘˜+1) and (π‘„π‘˜+1). Multiplying (3.14) by |𝑣|2π›½π‘˜+1βˆ’2𝑣 and integrating over Ξ©, we obtainπΆπ‘‘ξ€œπ‘‘π‘‘Ξ©|𝑣|2π›½π‘˜+1ξ€œπ‘‘π‘₯+𝐢Ω||βˆ‡π‘₯1𝑣||2+||π‘₯1||2𝑠||βˆ‡π‘₯2𝑣||2|𝑣|𝛽2π‘˜+1βˆ’2ξ€œπ‘‘π‘₯≀ℓΩ|𝑣|2π›½π‘˜+1𝑑π‘₯+𝑔′(𝑑),|𝑣|2π›½π‘˜+1βˆ’2𝑣.(3.25) Using the imbedding 𝑆10(Ξ©)β†ͺ𝐿2𝛽(Ξ©) once again, we get ξ€œΞ©ξ‚€||βˆ‡π‘₯1𝑣||2+||π‘₯1||2𝑠||βˆ‡π‘₯2𝑣||2|𝑣|𝛽2π‘˜+1ξ€œπ‘‘π‘₯β‰₯Ξ©|𝑣|2|𝑣|2𝛽2π‘˜+1βˆ’2=‖‖𝑣𝑑π‘₯π›½π‘˜+1β€–β€–2𝐿2𝛽(Ξ©)=ξ‚΅ξ€œΞ©|𝑣|2π›½π‘˜+2𝑑π‘₯1/𝛽.(3.26) Combining Holder’s and Young’s inequalities, we see that ξ€œΞ©π‘”ξ…ž(𝑑)|𝑣|2π›½π‘˜+1βˆ’2ξ‚΅ξ€œπ‘£π‘‘π‘₯≀Ω||π‘”ξ…ž||(𝑑)π‘šξ‚Άπ‘‘π‘₯1/π‘šξ‚΅ξ€œΞ©|𝑣|(2π›½π‘˜+1βˆ’1)𝑛𝑑π‘₯1/π‘›β‰€ξ€·βˆ«Ξ©||π‘”ξ…ž(||𝑑)π‘šξ€Έπ‘‘π‘₯π‘šβ€²/π‘šπ‘šξ…ž+ξ‚€βˆ«Ξ©|𝑣|(2π›½π‘˜+1βˆ’1)𝑛𝑑π‘₯𝑛′/π‘›π‘›ξ…ž,(3.27) where 1/π‘š+1/𝑛=1/π‘šξ…ž+1/π‘›ξ…ž=1. Choose 𝑛, π‘›ξ…ž such that ξ€·2π›½π‘˜+1ξ€Έβˆ’1𝑛=2π›½π‘˜+2,π‘›ξ…žπ‘›=1𝛽,(3.28) thus 𝑛=2π›½π‘˜+22π›½π‘˜+1βˆ’1,π‘›ξ…ž=2π›½π‘˜+12π›½π‘˜+1βˆ’1.(3.29) Hence π‘›π‘š==π‘›βˆ’12π›½π‘˜+1βˆ’12π›½π‘˜+2βˆ’2π›½π‘˜+1+1,π‘šξ…ž=2π›½π‘˜+1.(3.30) Then from (3.27), we infer that ξ€œΞ©π‘”ξ…ž(𝑑)|𝑣|2π›½π‘˜+1βˆ’21𝑣𝑑π‘₯β‰€π‘šξ…žβ€–β€–π‘”ξ…žβ€–β€–(𝑑)π‘šβ€²πΏπ‘š(Ξ©)+1π‘›ξ…žξ‚΅ξ€œΞ©|𝑣|2π›½π‘˜+2𝑑π‘₯1/𝛽.(3.31) Applying (3.26) and (3.31) in (3.25), we find that π‘‘ξ‚΅π‘’π‘‘π‘‘πœ†1π‘‘ξ€œΞ©|𝑣|2π›½π‘˜+1𝑑π‘₯+πΆπ‘’πœ†1π‘‘ξ‚΅ξ€œΞ©|𝑣|2π›½π‘˜+2𝑑π‘₯1/π›½β‰€πΆπ‘’πœ†1π‘‘ξ€œΞ©|𝑣|2π›½π‘˜+1𝑑π‘₯+πΆπ‘’πœ†1π‘‘β€–β€–π‘”ξ…žβ€–β€–(𝑑)π‘šβ€²πΏπ‘š(Ξ©).(3.32) Combining (π‘„π‘˜) and (3.32), using the uniform Gronwall inequality and taking into account assumption (3.18), we get (π‘ƒπ‘˜+1). On the other hand, integrating (3.32) from 𝑑 to 𝑑+1, we find (π‘„π‘˜+1). Now since 𝛽>1, and taking π‘˜β‰₯log𝛽𝑝/2, we get the desired estimate.

We will use the following lemma.

Lemma 3.5 (see [15]). If there exists 𝜎>0 such that βˆ«π‘‘βˆ’βˆžπ‘’πœŽπ‘ |πœ‘(𝑠)|2𝑑𝑠<∞, for all π‘‘βˆˆβ„, then lim𝛾→+βˆžξ€œπ‘‘βˆ’βˆžπ‘’βˆ’π›Ύ(π‘‘βˆ’π‘ )||||πœ‘(𝑠)2𝑑𝑠=0,π‘‘βˆˆβ„.(3.33)

Let π»π‘š=span{𝑒1,𝑒2,…,π‘’π‘š} in 𝐿2(Ξ©), and let π‘ƒπ‘šβˆΆπΏ2(Ξ©)β†’π»π‘š be the orthogonal projection, where {𝑒𝑖}βˆžπ‘–=1 are the eigenvectors of operator 𝐴=βˆ’πΊπ‘ . For any π‘’βˆˆπΏ2(Ξ©), we write𝑒=π‘ƒπ‘šξ€·π‘’+πΌβˆ’π‘ƒπ‘šξ€Έπ‘’=𝑒1+𝑒2.(3.34)

Lemma 3.6. For any π‘‘βˆˆβ„, any π΅βŠ‚πΏ2(Ξ©) and any πœ€, there exist 𝜏0(𝑑,𝐡,πœ€) and π‘š0βˆˆβ„• such that ||ξ€·πΌβˆ’π‘ƒπ‘šξ€Έπ‘£||22<πœ€,βˆ€πœβ‰€πœ0,βˆ€π‘’πœβˆˆπ΅,π‘šβ‰₯π‘š0.(3.35)

Proof. Multiplying (3.14) by 𝑣2=(πΌβˆ’π‘ƒπ‘š)𝑣 and then integrating over Ξ©, using |βˆ‡π‘£2|22β‰₯πœ†π‘š|𝑣2|22 and Cauchy’s inequality we get 𝑑||𝑣𝑑𝑑2||22+πœ†π‘š||𝑣2||22ξ€œβ‰€πΆΞ©||π‘“ξ…ž(||𝑒)𝑣2||𝑔𝑑π‘₯+πΆξ…ž(||𝑑)22.(3.36) We multiply (3.36) by π‘’πœ†π‘šπ‘‘ and use assumption (1.4). We get π‘‘ξ‚€π‘’π‘‘π‘‘πœ†π‘šπ‘‘||𝑣2||22ξ‚β‰€πΆπ‘’πœ†π‘šπ‘‘ξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2𝑑π‘₯+πΆπ‘’πœ†π‘šπ‘‘||π‘”ξ…ž||(𝑑)22.(3.37) Integrating (3.37) from 𝑠 to 𝑑, π‘’πœ†π‘šπ‘‘||𝑣2||(𝑑)22β‰€π‘’πœ†π‘šπ‘ ||𝑣2||(𝑠)22ξ€œ+πΆπ‘‘π‘ π‘’πœ†π‘šπ‘Ÿξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2ξ€œπ‘‘π‘₯π‘‘π‘Ÿ+πΆπ‘‘π‘ π‘’πœ†π‘šπ‘Ÿ||||𝑔′(π‘Ÿ)22π‘‘π‘Ÿβ‰€π‘’πœ†π‘šπ‘ ||||𝑣(𝑠)22ξ€œ+πΆπ‘‘βˆ’βˆžπ‘’πœ†π‘šπ‘Ÿξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2ξ€œπ‘‘π‘₯π‘‘π‘Ÿ+πΆπ‘‘βˆ’βˆžπ‘’πœ†π‘šπ‘Ÿ||π‘”ξ…ž||(π‘Ÿ)22π‘‘π‘Ÿ.(3.38) Now integrating (3.38) with respect to 𝑠 from 𝜏 to 𝑑, we infer that(π‘‘βˆ’πœ)π‘’πœ†π‘šπ‘‘||𝑣2||(𝑑)22β‰€ξ€œπ‘‘πœπ‘’πœ†π‘šπ‘Ÿ||||𝑣(π‘Ÿ)22ξ€œπ‘‘π‘Ÿ+𝐢(π‘‘βˆ’πœ)π‘‘βˆ’βˆžπ‘’πœ†π‘šπ‘Ÿξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2ξ€œπ‘‘π‘₯π‘‘π‘Ÿ+𝐢(π‘‘βˆ’πœ)π‘‘βˆ’βˆžπ‘’πœ†π‘šπ‘Ÿ||π‘”ξ…ž||(π‘Ÿ)22π‘‘π‘Ÿ.(3.39) Thus ||𝑣2||(𝑑)22≀1ξ€œπ‘‘βˆ’πœπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)||||𝑣(π‘Ÿ)22ξ€œπ‘‘π‘Ÿ+πΆπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)ξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2ξ€œπ‘‘π‘₯π‘‘π‘Ÿ+πΆπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)||π‘”ξ…ž||(π‘Ÿ)22π‘‘π‘Ÿ.(3.40) By Lemma 3.5 and since πœ†π‘šβ†’+∞ as π‘šβ†’+∞, there exist 𝜏1 and π‘š1 such that 1ξ€œπ‘‘βˆ’πœπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)πΆξ€œπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)||π‘”ξ…ž||(π‘Ÿ)22πœ€π‘‘π‘Ÿ<3,||||𝑣(π‘Ÿ)22πœ€π‘‘π‘Ÿ<3,(3.41) for all πœβ‰€πœ1 and π‘šβ‰₯π‘š1. For the second term of the right-hand side of (3.40), using Holder’s inequality we have ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)ξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2β‰€ξ€œπ‘‘π‘₯π‘‘π‘Ÿπ‘‘βˆ’βˆžξ‚΅ξ€œΞ©π‘’((βˆ’π‘βˆ’1)/(π‘βˆ’2)πœ†π‘š)(π‘‘βˆ’π‘Ÿ)|𝑒|2π‘βˆ’2𝑑π‘₯(π‘βˆ’2)/(π‘βˆ’1)ξ‚΅ξ€œΞ©π‘’βˆ’(π‘βˆ’1)πœ†π‘š(π‘‘βˆ’π‘Ÿ)|𝑣|2π‘βˆ’2𝑑π‘₯1/(π‘βˆ’1)β‰€ξ‚΅ξ€œπ‘‘π‘Ÿπ‘‘βˆ’βˆžπ‘’((βˆ’π‘βˆ’1)/(π‘βˆ’2)πœ†π‘š)(π‘‘βˆ’π‘Ÿ)‖𝑒‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)ξ‚Άπ‘‘π‘Ÿ(π‘βˆ’2)/(π‘βˆ’1)ξ‚΅ξ€œπ‘‘βˆ’βˆžπ‘’βˆ’(π‘βˆ’1)πœ†π‘š(π‘‘βˆ’π‘Ÿ)ξ€œπ‘‘π‘ŸΞ©|𝑣|2π‘βˆ’2𝑑π‘₯1/(π‘βˆ’1).(3.42) From Lemmas (3.5)–(3.7), we see that there exist 𝜏2 and π‘š2βˆˆβ„• such that πΆξ€œπ‘‘βˆ’βˆžπ‘’βˆ’πœ†π‘š(π‘‘βˆ’π‘Ÿ)ξ€œΞ©|𝑒|2(π‘βˆ’2)|𝑣|2πœ€π‘‘π‘₯π‘‘π‘Ÿ<3,βˆ€πœβ‰€πœ0,π‘šβ‰₯π‘š2.(3.43) Let 𝜏0=min{𝜏1,𝜏2} and π‘š0=max{π‘š1,π‘š2}, from (3.40), taking into account (3.41) and (3.43), we obtain (3.35).

Lemma 3.7 (see [9]). Let 𝐡 be a bounded subset in πΏπ‘ž(Ξ©)(π‘žβ‰₯1). If 𝐡 has a finite πœ€-net in πΏπ‘ž(Ξ©), then there exists an 𝑀=𝑀(𝐡,πœ€), such that for any π‘’βˆˆπ΅, the following estimate is valid: ξ€œΞ©(|𝑒|β‰₯𝑀)|𝑒|π‘žπ‘‘π‘₯<πœ€.(3.44)

Using Lemma 3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set {𝑒𝑑(𝑠)βˆΆπ‘ β‰€π‘‘,π‘’πœβˆˆπ΅} has a finite πœ–-net in 𝐿2(Ξ©). Therefore, we get the following result.

Lemma 3.8. For any π‘‘βˆˆβ„, any π΅βŠ‚πΏ2(Ξ©) that is bounded, and any πœ€>0, there exists 𝜏0≀𝑑 and 𝑀0>0 such thatξ€œΞ©(|𝑒|β‰₯𝑀)||𝑒𝑑||(𝑑)2𝑑π‘₯<πœ€,βˆ€πœ<𝜏0,𝑀>𝑀0,π‘’πœβˆˆπ΅.(3.45)

Lemma 3.9 (see [9]). For any π‘‘βˆˆβ„, any bounded set π΅βŠ‚πΏ2(Ξ©), and any πœ€>0, there exist 𝜏0 and 𝑀0>0 such that mes(Ξ©(𝑒(𝑑)β‰₯𝑀))<πœ€βˆ€πœβ‰€πœ0,𝑀β‰₯𝑀0,π‘’πœβˆˆπ΅,(3.46) where mes is the Lebesgue measure in ℝ𝑁 and Ξ©(𝑒(𝑑)β‰₯𝑀)={π‘₯βˆˆΞ©βˆΆπ‘’(𝑑,π‘₯)β‰₯𝑀}.

Lemma 3.10 (see [2]). Let {π‘ˆ(𝑑,𝜏)} be a norm-to-weak continuous process in 𝐿2(Ξ©) and πΏπ‘ž(Ξ©),π‘žβ‰₯2. Then {π‘ˆ(𝑑,𝜏)} is pullback asymptotically compact in πΏπ‘ž(Ξ©) if
(i){π‘ˆ(𝑑,𝜏)} is pullback asymptotically compact in 𝐿2(Ξ©); (ii)for any π‘‘βˆˆβ„, any bounded set π·βŠ‚πΏ2(Ξ©), and any πœ–>0, there exist 𝑀>0 and𝜏0≀𝑑supπœβ‰€πœ0supπ‘’πœβˆˆπ·ξ‚΅ξ€œΞ©(|π‘ˆ(𝑑,𝜏)π‘’πœ|β‰₯𝑀)||π‘ˆ(𝑑,𝜏)π‘’πœ||π‘žξ‚Άπ‘‘π‘₯β‰€πΆπœ–,(3.47) where 𝐢 is independent of 𝑀, 𝜏, π‘’πœ, and πœ–.

We are now ready to prove the existence of a pullback attractor in 𝐿2π‘βˆ’2(Ξ©).

Theorem 3.11. Assume that assumptions (1.3)–(1.7) and (3.18) hold. Then the process {π‘ˆ(𝑑,𝜏)} associated to problem (1.1) possesses a pullback attractor π’œ2π‘βˆ’2={𝐴2π‘βˆ’2(𝑑)}π‘‘βˆˆβ„ in 𝐿2π‘βˆ’2(Ξ©).

Proof. Because of Lemma 3.10, since {π‘ˆ(𝑑,𝜏)} has a pullback absorbing set in 𝐿2π‘βˆ’2(Ξ©), we only have to prove that for any π‘‘βˆˆβ„, any π΅βŠ‚πΏ2(Ξ©), and any πœ€>0, there exist 𝜏2≀𝑑 and 𝑀2>0 such that ξ€œΞ©(|𝑒|β‰₯𝑀)|𝑒|2π‘βˆ’2𝑑π‘₯β‰€πΆπœ€,βˆ€πœβ‰€πœ2,𝑀β‰₯𝑀2,π‘’πœβˆˆπ΅.(3.48) Taking the inner product of (1.1) with (π‘’βˆ’π‘€)+π‘βˆ’1 in 𝐿2(Ξ©), where (π‘’βˆ’π‘€)+=ξ‚»π‘’βˆ’π‘€if𝑒β‰₯𝑀,0if𝑒<𝑀,(3.49) we have ξ€œΞ©π‘’π‘‘(π‘’βˆ’π‘€)+π‘βˆ’1ξ€œπ‘‘π‘₯+(π‘βˆ’1)Ξ©ξ‚€||βˆ‡π‘₯1𝑒||2||π‘₯𝑑π‘₯+1||2𝑠||βˆ‡π‘₯2𝑒||2(π‘’βˆ’π‘€)+π‘βˆ’2+ξ€œπ‘‘π‘₯Ω𝑓(𝑒)(π‘’βˆ’π‘€)+π‘βˆ’1β‰€ξ€œπ‘‘π‘₯Ω𝑔(𝑑)(π‘’βˆ’π‘€)+π‘βˆ’1𝑑π‘₯.(3.50) Some standard computations give us ξ€œΞ©π‘“(𝑒)(π‘’βˆ’π‘€)+π‘βˆ’1ξ€œπ‘‘π‘₯β‰₯𝐢Ω(𝑒β‰₯𝑀)|𝑒|2π‘βˆ’2ξ€œπ‘‘π‘₯+𝐢Ω(𝑒β‰₯𝑀)|𝑒|𝑝𝑑π‘₯,(3.51)βˆ’ξ€œΞ©π‘’π‘‘(π‘’βˆ’π‘€)+π‘βˆ’1𝐢𝑑π‘₯≀4ξ€œΞ©(𝑒β‰₯𝑀)|𝑒|2π‘βˆ’21𝑑π‘₯+πΆξ€œΞ©(𝑒β‰₯𝑀)||𝑒𝑑||2𝑑π‘₯,(3.52)ξ€œΞ©π‘”(𝑑)(π‘’βˆ’π‘€)+π‘βˆ’1𝐢𝑑π‘₯≀4ξ€œΞ©(𝑒β‰₯𝑀)|𝑒|2π‘βˆ’21dπ‘₯+πΆξ€œΞ©(𝑒β‰₯𝑀)||||𝑔(𝑑)2𝑑π‘₯.(3.53) Combining (3.50)–(3.53), we find ξ€œΞ©(𝑒β‰₯𝑀)|𝑒|2π‘βˆ’2ξ‚΅ξ€œπ‘‘π‘₯≀𝐢Ω(𝑒β‰₯𝑀)||𝑒𝑑||2ξ€œπ‘‘π‘₯+Ξ©(𝑒β‰₯𝑀)||||𝑔(𝑑)2ξ€œπ‘‘π‘₯+𝐢Ω(𝑒β‰₯𝑀)|𝑒|𝑝.𝑑π‘₯(3.54) Applying Lemmas 3.7 and 3.8 to (3.54) we find there exist 𝜏0 and 𝑀0 such that ξ€œΞ©(𝑒β‰₯𝑀)|𝑒|2π‘βˆ’2𝑑π‘₯<πœ€βˆ€πœβ‰€πœ0,𝑀β‰₯𝑀0.(3.55) Repeating the above arguments with |(𝑒+𝑀)βˆ’|π‘βˆ’2(𝑒+𝑀)βˆ’ in place of (π‘’βˆ’π‘€)+π‘βˆ’1, we have ξ€œΞ©(π‘’β‰€βˆ’π‘€)|𝑒|2π‘βˆ’2𝑑π‘₯<πœ€βˆ€πœβ‰€πœ1,𝑀β‰₯𝑀1,(3.56) for some 𝜏1≀𝑑 and 𝑀1>0, where (𝑒+𝑀)βˆ’=𝑒+𝑀ifπ‘’β‰€βˆ’π‘€,0if𝑒>𝑀.(3.57) Letting 𝜏2=min{𝜏0,𝜏1} and 𝑀2=max{𝑀0,𝑀1} we have ξ€œΞ©(|𝑒|β‰₯𝑀2)|𝑒|2π‘βˆ’2𝑑π‘₯<πΆπœ€,βˆ€πœβ‰€πœ2,𝑀β‰₯𝑀2.(3.58) This completes the proof.

3.2. Existence of a Pullback Attractor in 𝑆20(Ξ©)

In this section, we prove the existence of a pullback attractor in 𝑆20(Ξ©).

Lemma 3.12. The process {π‘ˆ(𝑑,𝜏)} associated to (1.1) has a pullback absorbing set in 𝑆20(Ξ©).

Proof. We multiply (1.1) by βˆ’πΊπ‘ π‘’; then, using 𝑓(0)=0, we have ‖𝑒‖2𝑆20(Ξ©)=ξ€œΞ©π‘’π‘‘πΊπ‘ ξ€œπ‘’π‘‘π‘₯βˆ’Ξ©ξ‚€||βˆ‡π‘“β€²(𝑒)π‘₯1𝑒||2+||π‘₯1||2𝑠||βˆ‡π‘₯2𝑒||2ξ‚ξ€œπ‘‘π‘₯βˆ’Ξ©π‘”(𝑑)𝐺𝑠𝑒𝑑π‘₯.(3.59) Using 𝑓′(𝑒)β‰₯βˆ’β„“, Cauchy’s inequality, and argument as in Lemma 3.3, from (3.59) we have ‖𝑒‖2𝑆20(Ξ©)ξ‚€||𝑒≀2𝑑||22+ℓ‖𝑒‖2+||||𝑔(𝑑)22.(3.60) Taking into account (3.11), the proof is complete.

In order to prove the existence of the pullback attractor in 𝑆20(Ξ©), we will verify so-called β€œ(PDC) condition”, which is defined as follow

Definition 3.13. A process {π‘ˆ(𝑑,𝜏)} is said to satisfy (PDC) condition in 𝑋 if for any π‘‘βˆˆβ„, any bounded set π΅βŠ‚πΏ2(Ξ©) and any πœ€>0, there exists 𝜏0≀𝑑 and a finite dimensional subspace 𝑋1 of 𝑋 such that (i)⋃𝑃(πœβ‰€πœ0π‘ˆ(𝑑,𝜏)𝐡) is bounded in 𝑋; and(ii)β€–(πΌπ‘‹βˆ’π‘ƒ)π‘ˆ(𝑑,𝜏)π‘’πœβ€–π‘‹<πœ€, for all πœβ‰€πœ0 and π‘’πœβˆˆπ΅, where π‘ƒβˆΆπ‘‹β†’π‘‹1 is a canonical projection and 𝐼𝑋 is the identity.

Lemma 3.14 (see [13]). If a process {π‘ˆ(𝑑,𝜏)} satisfies (PDC) condition in 𝑋 then it is pullback asymptotically compact in 𝑋. Moreover, if 𝑋 is convex then the converse is true.

Lemma 3.15 (see [9]). Assume that 𝑓 satisfies (1.3) and (1.5). Then for any subset π΄βŠ‚πΏ2π‘βˆ’2(Ξ©), if πœ…(𝐴)<πœ€ in 𝐿2π‘βˆ’2(Ξ©), then we have πœ…(𝑓(𝐴))<πΆπœ€in𝐿2(Ξ©),(3.61) where the Kuratowski noncompactness measure πœ…(𝐡) in a Banach space 𝑋 defined as πœ…(𝐡)=inf{π›ΏβˆΆπ΅hasafinitecoveringbyballsin𝑋withradii𝛿}.(3.62)

Theorem 3.16. Assume that 𝑓 satisfies (1.3)–(1.5), 𝑔 satisfies (1.7) and (3.18). Then the process {π‘ˆ(𝑑,𝜏)} generated by (1.1) has a pullback attractor π’œ={𝐴(𝑑)βˆΆπ‘‘βˆˆβ„} in 𝑆20(Ξ©).

Proof. We consider a complete trajectory 𝑒(𝑑) lies on pullback attractor π’œ2π‘βˆ’2 in 𝐿2π‘βˆ’2(Ξ©) for π‘ˆ(𝑑,𝜏), that is, 𝑒(𝑑)βˆˆπ’œ2π‘βˆ’2(𝑑) and π‘ˆ(𝑑,𝜏)π‘’πœ=𝑒(𝑑), for all 𝑑β‰₯𝜏. Denoting 𝐴=βˆ’πΊπ‘  and multiplying (1.1) by 𝐴𝑒2=𝐴(πΌβˆ’π‘ƒπ‘š)𝑒=(πΌβˆ’π‘ƒπ‘š)𝐴𝑒 we have 𝑒𝑑,𝐴𝑒2ξ€Έ+‖‖𝑒2β€–β€–2𝑆20(Ξ©)+ξ€œΞ©π‘“(𝑒)𝐴𝑒2𝑑π‘₯=𝑔(𝑑),𝐴𝑒2ξ€Έ.(3.63) Using Holder’s inequality we get ‖‖𝑒2β€–β€–2𝑆20(Ξ©)ξ‚΅||≀𝐢(πΌβˆ’π‘ƒπ‘š)𝑒𝑑||22+||ξ€·πΌβˆ’π‘ƒπ‘šξ€Έ||𝑔(𝑑)22+ξ€œΞ©(𝑓(𝑒))2𝑑π‘₯.(3.64) Thanks to Lemmas 3.6 and 3.15 and the fact that π‘”βˆˆπΆloc(ℝ;𝐿2(Ξ©)), we see that {π‘ˆ(𝑑,𝜏)} satisfies condition (PDC) in 𝑆20(Ξ©). Now from Lemmas 3.3 and 3.14 we get the desired result.

4. Exponential Growth in 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©) of Pullback Attractors

In this section, we will give an exponential growth condition in 𝑆20(Ξ©)∩𝐿2π‘βˆ’2(Ξ©) for the pullback attractor π’œ(𝜏).

First, we recall a result in [17] which is necessary for the proof of our results.

Lemma 4.1. Let 𝑋, π‘Œ be Banach spaces such that 𝑋 is reflexive, and the inclusion π‘‹βŠ‚π‘Œ is continuous. Assume that {𝑒𝑛} is bounded sequence in 𝐿∞(𝑑0,𝑇;𝑋) such that 𝑒𝑛→𝑒 weakly in πΏπ‘ž(𝑑0,𝑇;𝑋) for some π‘žβˆˆ[1,+∞) and π‘’βˆˆπΆ0([𝑑0,𝑇];π‘Œ). Then, 𝑒(𝑑)βˆˆπ‘‹ for all π‘‘βˆˆ[𝑑0,𝑇] and ‖𝑒‖𝑋≀sup𝑛β‰₯1β€–β€–π‘’π‘›β€–β€–πΏβˆž(𝑑0,𝑇;𝑋)𝑑,βˆ€π‘‘βˆˆ0ξ€»,T.(4.1)

In the following theorem, instead of evaluating the functions 𝑒𝑛 which are differentiable enough and then using Lemma 4.1, we will formally evaluate the function 𝑒.

Theorem 4.2. Assume that 𝑓 satisfies (1.3)-(1.5), 𝑔 satisfies (H 2), (3.18) and the following conditions limπœβ†’βˆ’βˆžπ‘’πœ†1πœξ€œπœπœ+1||||𝑔′(𝑠)22𝑑𝑠=0,limπœβ†’βˆ’βˆžξ‚€π‘’πœ†1𝜏||||𝑔(𝑠)22=0.(4.2) Then π’œ(𝜏) satisfies limπœβ†’βˆ’βˆžπ‘’πœ†1πœξƒ―supπ‘€βˆˆπ’œ(𝜏)‖𝑀‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+supπ‘€βˆˆπ’œ(𝜏)‖𝑀‖2𝑆20(Ξ©)ξƒ°=0.(4.3)

Proof. We differentiate with respect to time in (1.1), then multiply by 𝑒𝑑, we get12𝑑||π‘’π‘‘π‘Ÿξ…ž||(π‘Ÿ)22+‖𝑒(π‘Ÿ)β€–2ξ€œ=βˆ’Ξ©ξ€œπ‘“β€²(𝑒)𝑒′(π‘Ÿ)𝑒′(π‘Ÿ)dπ‘₯+Ξ©||𝑒𝑔′(π‘Ÿ)𝑒′(π‘Ÿ)𝑑π‘₯β‰€β„“ξ…ž||(π‘Ÿ)22+12𝑒′(π‘Ÿ)22+12||||𝑔′(π‘Ÿ)22.(4.4) Integrating in the last inequality, in particular, we get ||π‘’ξ…ž||(π‘Ÿ)22≀||π‘’ξ…ž||(𝑠)22ξ€œ+(2β„“+1)π‘‘πœ+πœ–/2||π‘’ξ…ž||(πœƒ)22ξ€œπ‘‘πœƒ+π‘‘πœ+πœ–/2||π‘”ξ…ž||(πœƒ)22π‘‘πœƒ.(4.5) for all 𝜏+πœ–/2≀sβ‰€πœβ‰€π‘‘. Now, integrating with respect to 𝑠, between 𝜏+πœ–/2 and π‘Ÿξ‚€πœ–π‘Ÿβˆ’πœβˆ’2||π‘’ξ…ž||(π‘Ÿ)22β‰€πœ–ξ‚ƒξ‚€π‘Ÿβˆ’πœβˆ’2(ξ‚„ξ€œ2𝑙+1)+1π‘‘πœ+πœ–/2||π‘’ξ…ž||(πœƒ)22+ξ‚€πœ–π‘‘πœƒπ‘Ÿβˆ’πœβˆ’2ξ‚ξ€œπ‘‘πœ+πœ–/2||π‘”ξ…ž||(πœƒ)22π‘‘πœƒ.(4.6) for all 𝜏+πœ–/2β‰€π‘Ÿβ‰€π‘‘, in paricular, ||π‘’ξ…ž||(π‘Ÿ)22≀2πœ–βˆ’1πœ–ξ‚ƒξ‚€π‘Ÿβˆ’πœβˆ’2(ξ‚„ξ€œ2𝑙+1)+1π‘‘πœ+πœ–/2||π‘’ξ…ž||(πœƒ)22+ξ€œπ‘‘πœƒπ‘‘πœ+πœ–/2||π‘”ξ…ž||(πœƒ)22π‘‘πœƒ.(4.7) for all π‘Ÿβˆˆ[𝜏+πœ–,𝑑].
Multiplying (1.1) by 𝑒 and then integrating on Ξ©, we get12𝑑𝑑𝑑|𝑒|22+‖𝑒‖2+ξ€œΞ©ξ€œπ‘“(𝑒)𝑒𝑑π‘₯=Ξ©1𝑔(𝑑)𝑒𝑑π‘₯≀2πœ†1||||𝑔(𝑑)22+πœ†12|𝑒|22.(4.8) Using hypothesis (H1) and the fact that ‖𝑒‖2β‰₯(1/2)‖𝑒‖2+(πœ†1/2)|𝑒|22, we have 𝑑𝑑𝑑|𝑒|22+πœ†1|𝑒|22+2𝐢1|𝑒|π‘π‘βˆ’2𝐢2||Ξ©||≀1πœ†1||||𝑔(𝑑)22.(4.9) Integrating (4.9) from 𝜏 to π‘Ÿβˆˆ[𝜏,𝑑], we have ||||𝑒(π‘Ÿ)22+πœ†1ξ€œπ‘Ÿπœβ€–π‘’(𝑠)β€–2𝑑𝑠+2𝐢1ξ€œπ‘Ÿπœ||||𝑒(𝑠)𝑝𝑝||||𝑑𝑠≀𝑒(𝜏)22+1πœ†1ξ€œπ‘Ÿπœ||||𝑔(𝑠)22𝑑𝑠+2𝐢2||Ξ©||(π‘‘βˆ’πœ).(4.10) Thus, ||||𝑒(π‘Ÿ)22+ξ€œπ‘Ÿπœβ€–π‘’(𝑠)β€–2ξ€œπ‘‘π‘ +π‘Ÿπœ||||𝑒(𝑠)𝑝𝑝||||𝑑𝑠≀𝐢𝑒(𝜏)22+ξ€œπ‘Ÿπœ||||𝑔(𝑠)22𝑑𝑠+(π‘‘βˆ’πœ).(4.11) Multiplying (1.1) by 𝑒𝑑 then integrating over Ξ©, we have ||||𝑒′(π‘Ÿ)22+12π‘‘ξ‚΅β€–π‘‘π‘Ÿβ€–π‘’(π‘Ÿ)2ξ€œ+2Ω≀1𝐹(𝑒(π‘₯,π‘Ÿ))𝑑π‘₯2||||𝑔(π‘Ÿ)22+12||||𝑒′(π‘Ÿ)22.(4.12) Integrating now between π‘ βˆˆ[𝜏,π‘Ÿ] and π‘Ÿβ‰€π‘‘, we obtain ξ€œπ‘Ÿπ‘ ||||𝑒′(πœƒ)22π‘‘πœƒ+‖𝑒(π‘Ÿ)β€–2+ξ€œΞ©ξ€œπΉ(𝑒(π‘₯,π‘Ÿ))𝑑π‘₯β‰€π‘Ÿπ‘ ||||𝑔(πœƒ)22π‘‘πœƒ+‖𝑒(𝑠)β€–2+ξ€œΞ©πΉ(𝑒(π‘₯,𝑠))𝑑π‘₯.(4.13) From (4.13) and using hypothesis (H1), we get ξ€œπ‘Ÿπ‘ ||||𝑒′(πœƒ)22π‘‘πœƒ+‖𝑒(π‘Ÿ)β€–2+𝛼1ξ€œΞ©||||𝑒(π‘₯,π‘Ÿ)𝑝𝑑π‘₯βˆ’π›Ό2||Ξ©||≀‖𝑒(𝑠)β€–2+ξ€œπ‘Ÿπ‘ ||𝑔||(πœƒ)22π‘‘πœƒ+𝛼3ξ€œΞ©||𝑒||(π‘₯,π‘Ÿ)𝑝𝑑π‘₯+𝛼4||Ξ©||.(4.14) Hence ξ€œπ‘Ÿπ‘ ||||𝑒′(πœƒ)22π‘‘πœƒ+‖𝑒(π‘Ÿ)β€–2+𝛼1𝑒(π‘Ÿ)𝑝𝑝≀‖𝑒(𝑠)β€–2+ξ€œπ‘‘π‘ ||||𝑔(πœƒ)22π‘‘πœƒ+𝛼3||||𝑒(𝑠)𝑝𝑝+𝛼2+𝛼4ξ€Έ||Ξ©||.(4.15) Integrating inequality (4.15) with respect to 𝑠 from 𝜏 to π‘Ÿ, we obtain ‖(π‘Ÿβˆ’πœ)𝑒(π‘Ÿ)β€–2+||||𝑒(π‘Ÿ)π‘π‘ξ‚„ξ‚Έξ€œβ‰€πΆπ‘Ÿπœβ€–π‘’(𝑠)β€–2ξ€œπ‘‘π‘ +π‘Ÿπœ||||𝑒(𝑠)π‘π‘ξ‚Ήξ€œπ‘‘π‘ +𝐢(π‘‘βˆ’πœ)π‘‘πœ||𝑔||(𝑠)22||Ξ©||𝑑𝑠+𝐢(π‘‘βˆ’πœ),(4.16) for all 𝑑β‰₯𝜏, π‘Ÿβˆˆ[𝜏,𝑑].
From (4.16) and (4.11), we obtain that ‖𝑒(π‘Ÿ)β€–2+||||𝑒(π‘Ÿ)𝑝𝑝||π‘’β‰€πΆπœ||22+ξ€œπœπœ+2||||𝑔(𝑠)22ξ‚Ή,𝑑𝑠+1(4.17) for all π‘Ÿβˆˆ[𝜏+1,𝜏+2].
From (4.7), taking 𝑑=𝜏+3 and πœ–=2 we have||||𝑒′(π‘Ÿ)22ξ€œβ‰€(4𝑙+3)𝜏+3𝜏+1||||𝑒′(πœƒ)22ξ€œπ‘‘πœƒ+𝜏+3𝜏+1||||𝑔′(πœƒ)22π‘‘πœƒ(4.18) for all π‘Ÿβˆˆ[𝜏+2,𝜏+3].
Analogously, and if we take 𝑠=𝜏+1 and π‘Ÿ=𝑑=𝜏+3 in inequlity (4.15), thenξ€œπœ+3𝜏+1||||𝑒′(𝑠)22‖𝑑𝑠+‖𝑒(𝜏+3)2+𝛼1||||𝑒(𝜏+3)𝑝𝑝≀‖𝑒(𝜏+1)β€–2+ξ€œπœπœ+3||||𝑔(𝑠)22𝑑𝑠+𝛼3||||𝑒(𝜏+1)𝑝𝑝+𝛼2+𝛼4ξ€Έ||Ξ©||.(4.19) From (4.18) and (4.19), we obtain ||π‘’ξ…ž||(π‘Ÿ)22‖≀(4𝑙+3)𝑒(𝜏+1)β€–2+𝛼3||||𝑒(𝜏+1)𝑝𝑝𝛼+(4𝑙+3)2+𝛼4ξ€Έ||Ξ©||+ξ€œπœπœ+3||||𝑔(𝑠)22ξ‚Ή+ξ€œπ‘‘π‘ πœ+3𝜏+1||π‘”ξ…ž||(πœƒ)22π‘‘πœƒ(4.20) for all π‘Ÿβˆˆ[𝜏+2,𝜏+3].
Owing to this inequality and (4.17), we have||||𝑒′(π‘Ÿ)22ξ‚Έ||π‘’β‰€πΆπœ||22+ξ€œπœπœ+3||||𝑔(𝑠)22+||π‘”ξ…ž||(𝑠)22ξ‚„ξ‚Ή,𝑑𝑠+1(4.21) for all π‘Ÿβˆˆ[𝜏+2,𝜏+3].
From (3.23), (3.60) and using Young’s inequality, we have‖𝑒(π‘Ÿ)‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+‖𝑒(π‘Ÿ)β€–2𝑆20(Ξ©)ξ‚€||π‘’β‰€πΆξ…ž||(π‘Ÿ)22+‖𝑒(π‘Ÿ)β€–2+||||𝑒(π‘Ÿ)𝑝𝑝||||+1+𝑔(π‘Ÿ)22,(4.22) for all π‘Ÿβ‰₯𝜏.
From (4.22) and thank to (4.21), we have‖𝑒(π‘Ÿ)‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+‖𝑒(π‘Ÿ)β€–2𝑆20(Ξ©)ξ‚Έ||π‘’β‰€πΆπœ||22+ξ€œπœπœ+3||||𝑔(𝑠)22+||||𝑔′(𝑠)22𝑑𝑠+1+𝐢‖𝑒(π‘Ÿ)β€–2+||||𝑒(π‘Ÿ)𝑝𝑝+||||𝑔(π‘Ÿ)22(4.23) for all π‘Ÿβˆˆ[𝜏+2,𝜏+3]. From (4.23) and thanks to (4.17), we have ‖𝑒(π‘Ÿ)‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+‖𝑒(π‘Ÿ)β€–2𝑆20(Ξ©)||π‘’β‰€πΆπœ||+ξ€œπœπœ+3||||𝑔(𝑠)22+||||𝑔′(𝑠)22𝑑𝑠+1+sup[]π‘Ÿβˆˆπœ+2,𝜏+3||||𝑔(π‘Ÿ)22ξƒ­,(4.24) for all π‘Ÿβˆˆ[𝜏+2,𝜏+3]. Now, observe that by Cauchy’s inequality ||||β‰₯||||+ξ‚΅ξ€œπ‘”(π‘Ÿ)𝑔(𝜏+2)𝜏+3𝜏+2||||𝑔′(𝑠)22𝑑𝑠1/2,(4.25) for all π‘Ÿβˆˆ[𝜏+2,𝜏+3], πœβˆˆβ„,π‘’πœβˆˆπΏ2(Ξ©).
Thus, from (4.23), we haveβ€–β€–π‘ˆ(𝜏+2,𝜏)π‘’πœβ€–β€–πΏ2π‘βˆ’22π‘βˆ’2(Ξ©)+β€–β€–π‘ˆ(𝜏+2,𝜏)π‘’πœβ€–β€–2𝑆20(Ξ©)ξ‚Έ||π‘’β‰€πΆπœ||22+ξ€œπœπœ+3ξ‚€||||𝑔(𝑠)22+||||𝑔′(𝑠)22||||𝑑𝑠+1+𝑔(𝜏+2)22ξ‚Ή.(4.26) for all πœβˆˆβ„,β€‰π‘’πœβˆˆπΏ2(Ξ©). From this inequlity, and the fact that π’œ(𝜏)=π‘ˆ(𝜏,πœβˆ’2)𝐴(πœβˆ’2), we obtain ‖𝑣‖𝐿2π‘βˆ’22π‘βˆ’2(Ξ©)+‖𝑣‖2𝑆20(Ξ©)≀𝐢supπ‘€βˆˆπ΄(πœβˆ’2)|𝑀|22+ξ€œπœ+1πœβˆ’2ξ‚€||||𝑔(𝑠)22+||||𝑔′(𝑠)22||||𝑑𝑠+1+𝑔(𝜏)22ξƒ­.(4.27) for all π‘£βˆˆπ΄(𝜏), and any πœβˆˆβ„. Now, thank to (4.2), (4.3), we obtain (4.3) from (4.27).

Acknowledgment

The authors thanks the reviewers very much for valuable comments and suggestions.

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