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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 172956, 15 pages
Research Article

Global Attractor of Atmospheric Circulation Equations with Humidity Effect

College of Mathematics and Software Science, Sichuan Normal University, Sichuan, Chengdu 610066, China

Received 1 June 2012; Accepted 15 July 2012

Academic Editor: JinhuΒ LΓΌ

Copyright Β© 2012 Hong Luo. 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.


Global attractor of atmospheric circulation equations is considered in this paper. Firstly, it is proved that this system possesses a unique global weak solution in 𝐿2(Ξ©,𝑅4). Secondly, by using C-condition, it is obtained that atmospheric circulation equations have a global attractor in 𝐿2(Ξ©,𝑅4).

1. Introduction

This paper is concerned with global attractor of the following initial-boundary problem of atmospheric circulation equations involving unknown functions (𝑒,𝑇,π‘ž,𝑝) at (π‘₯,𝑑)=(π‘₯1,π‘₯2,𝑑)βˆˆΞ©Γ—(0,∞) (Ξ©=(0,2πœ‹)Γ—(0,1) is a period of 𝐢∞ field (βˆ’βˆž,+∞)Γ—(0,1)): πœ•π‘’πœ•π‘‘=π‘ƒπ‘Ÿ(Ξ”π‘’βˆ’βˆ‡π‘βˆ’πœŽπ‘’)+π‘ƒπ‘Ÿξ‚€ξ‚ξ‚π‘…π‘‡βˆ’π‘…π‘žβ†’πœ…βˆ’(π‘’β‹…βˆ‡)𝑒,(1.1)πœ•π‘‡πœ•π‘‘=Δ𝑇+𝑒2βˆ’(π‘’β‹…βˆ‡)𝑇+𝑄,(1.2)πœ•π‘žπœ•π‘‘=πΏπ‘’Ξ”π‘ž+𝑒2βˆ’(π‘’β‹…βˆ‡)π‘ž+𝐺,(1.3)div𝑒=0,(1.4) where π‘ƒπ‘Ÿ, 𝑅, 𝑅, and 𝐿𝑒 are constants, 𝑒=(𝑒1,𝑒2), 𝑇, π‘ž, and 𝑝 denote velocity field, temperature, humidity, and pressure, respectively; 𝑄, 𝐺 are known functions, and 𝜎 is constant matrix: βŽ›βŽœβŽœβŽπœŽπœŽ=0πœ”πœ”πœŽ1⎞⎟⎟⎠.(1.5)

The problems (1.1)–(1.4) are supplemented with the following Dirichlet boundary condition at π‘₯2=0,1 and periodic condition for π‘₯1: (𝑒,𝑇,π‘ž)=0,π‘₯2ξ€·=0,1,(𝑒,𝑇,π‘ž)0,π‘₯2ξ€Έξ€·=(𝑒,𝑇,π‘ž)2πœ‹,π‘₯2ξ€Έ,(1.6) and initial value conditions 𝑒(𝑒,𝑇,π‘ž)=0,𝑇0,π‘ž0ξ€Έ,𝑑=0.(1.7)

The partial differential equations (1.1)–(1.7) were firstly presented in atmospheric circulation with humidity effect [1]. Atmospheric circulation is one of the main factors affecting the global climate, so it is very necessary to understand and master its mysteries and laws. Atmospheric circulation is an important mechanism to complete the transports and balance of atmospheric heat and moisture and the conversion between various energies. On the contrary, it is also the important result of these physical transports, balance and conversion. Thus, it is of necessity to study the characteristics, formation, preservation, change and effects of the atmospheric circulation and master its evolution law, which is not only the essential part of human’s understanding of nature, but also the helpful method of changing and improving the accuracy of weather forecasts, exploring global climate change, and making effective use of climate resources.

The atmosphere and ocean around the earth are rotating geophysical fluids, which are also two important components of the climate system. The phenomena of the atmosphere and ocean are extremely rich in their organization and complexity, and a lot of them cannot be produced by laboratory experiments. The atmosphere or the ocean or the couple atmosphere and ocean can be viewed as an initial and boundary value problem [2–5], or an infinite dimensional dynamical system [6–8]. We deduce the atmospheric circulation model (1.1)–(1.7) which is able to show features of atmospheric circulation and is easy to be studied from the very complex atmospheric circulation model based on the actual background and meteorological data, and we present global solutions of atmospheric circulation equations with the use of the 𝑇-weakly continuous operator [1]. In fact, there are numerous papers on this topic [9–13]. Compared with some similar papers, we add humidity function in this paper. We propose firstly the atmospheric circulation equation with humidity function which does not appear in the previous literature.

As far as the theory of infinite-dimensional dynamical system is concerned, we refer to [9–11, 14–18]. In the study of infinite dimensional dynamical system, the long-time behavior of the solution to equations is an important issue. The long-time behavior of the solution to equations can be shown by the global attractor with the finite-dimensional characteristics. Some authors have already studied the existence of the global attractor for some evolution equations [2, 3, 13, 19–21]. The global attractor strictly defined as πœ”-limit set of ball, which under additional assumptions is nonempty, compact, and invariant [13, 17]. Attractor theory has been intensively investigated within the science, mathematics, and engineering communities. LΓΌ et al. [22–25] apply the current theoretical results or approaches to investigate the global attractor of complex multiscroll chaotic systems. We obtain existence of global attractor for the atmospheric circulation equations from the mathematical perspective in this paper.

The paper is organized as follows. In Section 2, we recall preliminary results. In Section 3, we present uniqueness of the solution to the atmospheric circulation equations. In Section 4, we obtain global attractor of the equations.

‖⋅‖𝑋 denote norm of the space 𝑋; 𝐢 and 𝐢𝑖 are variable constants. Let 𝐻={πœ™=(𝑒,𝑇,π‘ž)∈𝐿2(Ξ©,𝑅4)βˆ£πœ™ satisfy (1.4), (1.6)}, and 𝐻1={πœ™=(𝑒,𝑇,π‘ž)∈𝐻1(Ξ©,𝑅4)βˆ£πœ™ satisfy (1.4), (1.6)}.

2. Preliminaries

Let 𝑋 and 𝑋1 be two Banach spaces, 𝑋1βŠ‚π‘‹ a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on 𝑋, given by 𝑑𝑒𝑑𝑑=𝐿𝑒+𝐺(𝑒),𝑒(π‘₯,0)=𝑒0,(2.1) where 𝑒(𝑑) is an unknown function, πΏβˆΆπ‘‹1→𝑋 a linear operator, and πΊβˆΆπ‘‹1→𝑋 a nonlinear operator.

A family of operators 𝑆(𝑑)βˆΆπ‘‹β†’π‘‹ (𝑑β‰₯0) is called a semigroup generated by (2.1) if it satisfies the following properties:(1)𝑆(𝑑)βˆΆπ‘‹β†’π‘‹ is a continuous map for any 𝑑β‰₯0;(2)𝑆(0)=π‘–π‘‘βˆΆπ‘‹β†’π‘‹ is the identity;(3)𝑆(𝑑+𝑠)=𝑆(𝑑)⋅𝑆(𝑠), for all 𝑑,𝑠β‰₯0. Then, the solution of (2.1) can be expressed as 𝑒𝑑,𝑒0ξ€Έ=𝑆(𝑑)𝑒0.(2.2)Next, we introduce the concepts and definitions of invariant sets, global attractors, and πœ”-limit sets for the semigroup 𝑆(𝑑).

Definition 2.1. Let 𝑆(𝑑) be a semigroup defined on 𝑋. A set Ξ£βŠ‚π‘‹ is called an invariant set of 𝑆(𝑑) if 𝑆(𝑑)Ξ£=Ξ£, for all 𝑑β‰₯0. An invariant set Ξ£ is an attractor of 𝑆(𝑑) if Ξ£ is compact, and there exists a neighborhood π‘ˆβŠ‚π‘‹ of Ξ£ such that for any 𝑒0βˆˆπ‘ˆ, infπ‘£βˆˆΞ£β€–β€–π‘†(𝑑)𝑒0β€–β€–βˆ’π‘£π‘‹βŸΆ0,asπ‘‘βŸΆβˆž.(2.3)
In this case, we say that Ξ£ attracts π‘ˆ. Particularly, if Ξ£ attracts any bounded set of 𝑋, Ξ£ is called a global attractor of 𝑆(𝑑) in 𝑋.
For a set π·βŠ‚π‘‹, we define the πœ”-limit set of 𝐷 as follows: ξ™πœ”(𝐷)=𝑠β‰₯0ξšπ‘‘β‰₯𝑠𝑆(𝑑)𝐷,(2.4) where the closure is taken in the 𝑋-norm. Lemma 2.2 is the classical existence theorem of global attractor by Temam [13].

Lemma 2.2. Let 𝑆(𝑑)βˆΆπ‘‹β†’π‘‹ be the semigroup generated by (2.1). Assume that the following conditions hold:(1)𝑆(𝑑) has a bounded absorbing set π΅βŠ‚π‘‹, that is, for any bounded set π΄βŠ‚π‘‹ there exists a time 𝑑𝐴β‰₯0 such that 𝑆(𝑑)𝑒0∈𝐡,  for  all  𝑒0∈𝐴 and 𝑑>𝑑𝐴;(2)𝑆(𝑑) is uniformly compact, that is, for any bounded set π‘ˆβŠ‚π‘‹ and some 𝑇>0 sufficiently large, the set ⋃𝑑β‰₯𝑇𝑆(𝑑)π‘ˆ is compact in 𝑋.
Then the πœ”-limit set π’œ=πœ”(𝐡) of 𝐡 is a global attractor of (2.1), and π’œ is connected providing 𝐡 is connected.

Definition 2.3 (see [19]). We say that 𝑆(𝑑)βˆΆπ‘‹β†’π‘‹ satisfies 𝐢-condition, if for any bounded set π΅βŠ‚π‘‹ and πœ€>0, there exist 𝑑𝐡>0 and a finite dimensional subspace 𝑋1βŠ‚π‘‹ such that {𝑃𝑆(𝑑)𝐡} is bounded, and β€–(πΌβˆ’π‘ƒ)𝑆(𝑑)𝑒‖𝑋<πœ€,βˆ€π‘‘β‰₯𝑑𝐡,π‘’βˆˆπ΅,(2.5) where π‘ƒβˆΆπ‘‹β†’π‘‹1 is a projection.

Lemma 2.4 (see [19]). Let 𝑆(𝑑)βˆΆπ‘‹β†’π‘‹ (𝑑β‰₯0) be a dynamical systems. If the following conditions are satisfied:(1)there exists a bounded absorbing set π΅βŠ‚π‘‹;(2)𝑆(𝑑) satisfies 𝐢-condition,
then 𝑆(𝑑) has a global attractor in 𝑋.

From Linear elliptic equation theory, one has the following.

Lemma 2.5. The eigenvalue equation: ξ€·π‘₯βˆ’Ξ”π‘‡1,π‘₯2ξ€Έξ€·π‘₯=𝛽𝑇1,π‘₯2ξ€Έ,ξ€·π‘₯1,π‘₯2ξ€Έβˆˆ(0,2πœ‹)Γ—(0,1),𝑇=0,π‘₯2𝑇=0,1,0,π‘₯2ξ€Έξ€·=𝑇2πœ‹,π‘₯2ξ€Έ(2.6) has eigenvalue {π›½π‘˜}βˆžπ‘˜=1, and 0<𝛽1≀𝛽2≀⋯,π›½π‘˜βŸΆβˆž,asπ‘˜βŸΆβˆž.(2.7)

3. Uniqueness of Global Solution

Theorem 3.1. If ξ‚πœŽπ›½1β‰₯max{(𝑅+1)2,((π‘…βˆ’1)2/𝐿𝑒)}, and 𝛽1 is the first eigenvalue of elliptic equation (2.6), then the weak solution to (1.1)–(1.7) is unique.

Proof. From [1], (𝑒,𝑇,π‘ž)∈𝐿∞((0,𝑇),𝐻)∩𝐿2((0,𝑇),𝐻1), 0<𝑇<∞ is the weak solution to (1.1)–(1.7). Then for all (𝑣,𝑆,𝑧)∈𝐻1, 0≀𝑑≀𝑇, we have 1π‘ƒπ‘Ÿξ€œΞ©ξ€œπ‘’π‘£π‘‘π‘₯+Ξ©ξ€œπ‘‡π‘†π‘‘π‘₯+Ξ©ξ€œπ‘žπ‘§π‘‘π‘₯=𝑑0ξ€œΞ©ξ‚ƒξ‚€ξ‚ξ‚π‘£βˆ’βˆ‡π‘’βˆ‡π‘£βˆ’πœŽπ‘’π‘£+π‘…π‘‡βˆ’π‘…π‘ž2βˆ’1π‘ƒπ‘Ÿ(π‘’β‹…βˆ‡)π‘’π‘£βˆ’βˆ‡π‘‡βˆ‡π‘†+𝑒2π‘†βˆ’(π‘’β‹…βˆ‡)𝑇𝑆+π‘„π‘†βˆ’πΏπ‘’βˆ‡π‘žβˆ‡π‘§+𝑒2ξ‚„+1π‘§βˆ’(π‘’β‹…βˆ‡)π‘žπ‘§+𝐺𝑧𝑑π‘₯π‘‘π‘‘π‘ƒπ‘Ÿξ€œΞ©π‘’0ξ€œπ‘£π‘‘π‘₯+Ω𝑇0ξ€œπ‘†π‘‘π‘₯+Ξ©π‘ž0𝑧𝑑π‘₯.(3.1)
Set (𝑒1,𝑇1,π‘ž1) and (𝑒2,𝑇2,π‘ž2) are two weak solutions to (1.1)–(1.7), which satisfy (3.1). Let (𝑒,𝑇,π‘ž)=(𝑒1,𝑇1,π‘ž1)βˆ’(𝑒2,𝑇2,π‘ž2). Then, 1π‘ƒπ‘Ÿξ€œΞ©ξ€œπ‘’π‘£π‘‘π‘₯+Ξ©ξ€œπ‘‡π‘†π‘‘π‘₯+Ξ©ξ€œπ‘žπ‘§π‘‘π‘₯=𝑑0ξ€œΞ©ξ‚ƒξ‚€ξ‚ξ‚π‘£βˆ’βˆ‡π‘’βˆ‡π‘£βˆ’πœŽπ‘’π‘£+π‘…π‘‡βˆ’π‘…π‘ž2+1π‘ƒπ‘Ÿξ€·π‘’2ξ€Έπ‘’β‹…βˆ‡21π‘£βˆ’π‘ƒπ‘Ÿξ€·π‘’1ξ€Έπ‘’β‹…βˆ‡1π‘£βˆ’βˆ‡π‘‡βˆ‡π‘†+𝑒2𝑒𝑆+2ξ€Έπ‘‡β‹…βˆ‡2ξ€·π‘’π‘†βˆ’1ξ€Έπ‘‡β‹…βˆ‡1π‘†βˆ’πΏπ‘’βˆ‡π‘žβˆ‡π‘§+𝑒2𝑒𝑧+2ξ€Έπ‘žβ‹…βˆ‡2π‘§βˆ’ξ€·π‘’1ξ€Έπ‘žβ‹…βˆ‡1𝑧𝑑π‘₯𝑑𝑑.(3.2)
Let (𝑣,𝑆,𝑧)=(𝑒,𝑇,π‘ž). We obtain from (3.2) the following: 1π‘ƒπ‘Ÿξ€œΞ©|𝑒|2ξ€œπ‘‘π‘₯+Ξ©||𝑇||2ξ€œπ‘‘π‘₯+Ξ©||π‘ž||2=ξ€œπ‘‘π‘₯𝑑0ξ€œΞ©ξ‚Έβˆ’||||βˆ‡π‘’2ξ‚€ξ‚ξ‚π‘’βˆ’πœŽπ‘’β‹…π‘’+π‘…π‘‡βˆ’π‘…π‘ž2+1π‘ƒπ‘Ÿξ€·π‘’2ξ€Έπ‘’β‹…βˆ‡21π‘’βˆ’π‘ƒπ‘Ÿξ€·π‘’1ξ€Έπ‘’β‹…βˆ‡1π‘’βˆ’||||βˆ‡π‘‡2+𝑒2𝑒𝑇+2ξ€Έπ‘‡β‹…βˆ‡2ξ€·π‘’π‘‡βˆ’1ξ€Έπ‘‡β‹…βˆ‡1π‘‡βˆ’πΏπ‘’||||βˆ‡π‘ž2+𝑒2π‘ž+𝑒2ξ€Έπ‘žβ‹…βˆ‡2ξ€·π‘’π‘žβˆ’1ξ€Έπ‘žβ‹…βˆ‡1π‘žξ‚Ήβ‰€ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξ€œΞ©ξ‚ƒβˆ’||||βˆ‡π‘’2βˆ’||||βˆ‡π‘‡2βˆ’πΏπ‘’||||βˆ‡π‘ž2ξ‚„+ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξ€œΞ©ξ‚ƒβˆ’ξ‚πœŽ|𝑒|2+(𝑅+1)𝑇𝑒2βˆ’ξ‚€ξ‚ξ‚π‘…βˆ’1π‘žπ‘’2ξ‚„+ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξ€œΞ©ξ‚Έ1π‘ƒπ‘Ÿ(π‘’β‹…βˆ‡)𝑒2𝑒+(π‘’β‹…βˆ‡)𝑇2𝑇+(π‘’β‹…βˆ‡)π‘ž2π‘žξ‚Ήβ‰€ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξ€œΞ©ξ‚ƒβˆ’||||βˆ‡π‘’2βˆ’||||βˆ‡π‘‡2βˆ’πΏπ‘’||||βˆ‡π‘ž2ξ‚„+ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξ€œΞ©βŽ‘βŽ’βŽ’βŽ’βŽ£βˆ’ξ‚πœŽ|𝑒|2||𝑒+ξ‚πœŽ2||2+(𝑅+1)2||𝑇||2ξ‚πœŽ2+ξ‚€ξ‚ξ‚π‘…βˆ’12||π‘ž||2ξ‚πœŽ2⎀βŽ₯βŽ₯βŽ₯⎦+ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξƒ¬βˆš2π‘ƒπ‘Ÿβ€–π‘’β€–πΏ2β€–β€–βˆ‡π‘’2‖‖𝐿2β€–βˆ‡π‘’β€–πΏ2+√2‖𝑒‖𝐿1/22β€–βˆ‡π‘’β€–πΏ1/22β€–β€–βˆ‡π‘‡2‖‖𝐿2‖𝑇‖𝐿1/22β€–βˆ‡π‘‡β€–πΏ1/22+√2‖𝑒‖𝐿1/22β€–βˆ‡π‘’β€–πΏ1/22β€–β€–βˆ‡π‘ž2β€–β€–β€–π‘žβ€–πΏ1/22β€–βˆ‡π‘žβ€–πΏ1/22ξƒ­β‰€ξ€œπ‘‘π‘‘π‘‘0ξ€œΞ©ξ‚Έβˆ’||||βˆ‡π‘’2βˆ’12||||βˆ‡π‘‡2βˆ’πΏπ‘’2||||βˆ‡π‘ž2ξ‚Ή+ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξƒ¬βˆš2π‘ƒπ‘Ÿβ€–π‘’β€–πΏ2β€–β€–βˆ‡π‘’2‖‖𝐿2β€–βˆ‡π‘’β€–πΏ2+√2‖𝑒‖𝐿2β€–β€–β€–βˆ‡π‘’β€–βˆ‡π‘‡2‖‖𝐿2+√2β€–β€–βˆ‡π‘‡2‖‖𝐿2‖𝑇‖𝐿2β€–βˆ‡π‘‡β€–πΏ2+√2‖𝑒‖𝐿2β€–βˆ‡π‘’β€–πΏ2β€–β€–βˆ‡π‘ž2‖‖𝐿2+√2β€–π‘žβ€–πΏ2β€–βˆ‡π‘žβ€–πΏ2β€–β€–βˆ‡π‘ž2‖‖𝐿2ξƒ­β‰€ξ€œπ‘‘π‘‘π‘‘0ξ€œΞ©ξ‚Έβˆ’||||βˆ‡π‘’2βˆ’12||||βˆ‡π‘‡2βˆ’πΏπ‘’2||||βˆ‡π‘ž2ξ‚Ή+ξ€œπ‘‘π‘₯𝑑𝑑𝑑0ξ‚Έβ€–βˆ‡π‘’β€–2𝐿2+3𝑃2π‘Ÿβ€–π‘’β€–2𝐿2β€–β€–βˆ‡π‘’2β€–β€–2𝐿2+3‖𝑒‖2𝐿2β€–β€–βˆ‡π‘‡2β€–β€–2𝐿2+3‖𝑒‖2𝐿2β€–β€–βˆ‡π‘ž2β€–β€–2𝐿2+12β€–βˆ‡π‘‡β€–2𝐿2+β€–β€–βˆ‡π‘‡2β€–β€–2𝐿2‖𝑇‖2𝐿2+𝐿𝑒2β€–βˆ‡π‘žβ€–2𝐿2+2πΏπ‘’β€–π‘žβ€–2𝐿2||βˆ‡π‘ž2||2𝐿2ξ‚Ήβ‰€ξ€œπ‘‘π‘‘π‘‘0ξ‚Έ3𝑃2π‘Ÿβ€–π‘’β€–2𝐿2β€–β€–βˆ‡π‘’2β€–β€–2𝐿2+3‖𝑒‖2𝐿2β€–β€–βˆ‡π‘‡2β€–β€–2𝐿2+3‖𝑒‖2𝐿2β€–β€–βˆ‡π‘ž2β€–β€–2𝐿2+β€–β€–βˆ‡π‘‡2β€–β€–2𝐿2‖𝑇‖2𝐿2+2πΏπ‘’β€–π‘žβ€–2𝐿2β€–β€–βˆ‡π‘ž2β€–β€–2𝐿2𝑑𝑑.(3.3) Then, ‖𝑒‖2𝐿2+‖𝑇‖2𝐿2+β€–π‘žβ€–2𝐿2ξ€œβ‰€πΆπ‘‘0‖𝑒‖2𝐿2+‖𝑇‖2𝐿2+β€–π‘žβ€–2𝐿2β€–β€–ξ‚ξ‚€βˆ‡π‘’2β€–β€–2𝐿2+β€–β€–βˆ‡π‘‡2β€–β€–2𝐿2+β€–β€–βˆ‡π‘ž2β€–β€–2𝐿2ξ€œξ‚ξ‚„π‘‘π‘‘β‰€πΆπ‘‘0‖𝑒‖2𝐿2+‖𝑇‖2𝐿2+β€–π‘žβ€–2𝐿2𝑑𝑑.(3.4)
By using the Gronwall inequality, it follows that ‖𝑒‖2𝐿2+‖𝑇‖2𝐿2+β€–π‘žβ€–2𝐿2≀0,(3.5) which imply (𝑒,𝑇,π‘ž)≑0. Thus, the weak solution to (1.1)–(1.7) is unique.

4. Existence of Global Attractor

Theorem 4.1. If ξ‚πœŽπ›½1β‰₯max{(𝑅+1)2,(π‘…βˆ’1)2/𝐿𝑒}, and 𝛽1 is the first eigenvalue of elliptic equation (2.6), then (1.1)–(1.7) have a global attractor in 𝐿2(Ξ©,𝑅4).

Proof. According to Lemma 2.4, we prove Theorem 4.1 in the following two steps.
Step  1. Equations (1.1)–(1.7) have an absorbing set in 𝐻.
Multiply (1.1) by 𝑒 and integrate the product in Ξ©: 1π‘ƒπ‘Ÿξ€œΞ©π‘‘π‘’ξ€œπ‘‘π‘‘π‘’π‘‘π‘₯=Ω1Ξ”π‘’βˆ’βˆ‡π‘βˆ’πœŽπ‘’+π‘…π‘‡βˆ’π‘…π‘žξ‚πœ…βˆ’π‘ƒπ‘Ÿξ‚Ή(π‘’β‹…βˆ‡)𝑒𝑒𝑑π‘₯.(4.1) Then, 12π‘ƒπ‘Ÿπ‘‘ξ€œπ‘‘π‘‘Ξ©π‘’2ξ€œπ‘‘π‘₯=Ξ©ξ‚ƒβˆ’||||βˆ‡π‘’2ξ‚€ξ‚ξ‚π‘’βˆ’πœŽπ‘’β‹…π‘’+π‘…π‘‡βˆ’π‘…π‘ž2𝑑π‘₯.(4.2)
Multiply (1.2) by 𝑇 and integrate the product in Ξ©: ξ€œΞ©π‘‘π‘‡ξ€œπ‘‘π‘‘π‘‡π‘‘π‘₯=ΩΔ𝑇+𝑒2ξ€»βˆ’(π‘’β‹…βˆ‡)𝑇+𝑄𝑇𝑑π‘₯.(4.3) Then, 12π‘‘ξ€œπ‘‘π‘‘Ξ©π‘‡2ξ€œπ‘‘π‘₯=Ξ©ξ‚€βˆ’||||βˆ‡π‘‡2+𝑒2𝑇𝑑π‘₯+𝑄𝑇𝑑π‘₯.(4.4)
Multiply (1.3) by π‘ž and integrate the product in Ξ©: ξ€œΞ©π‘‘π‘žξ€œπ‘‘π‘‘π‘žπ‘‘π‘₯=Ξ©ξ€ΊπΏπ‘’Ξ”π‘ž+𝑒2ξ€»βˆ’(π‘’β‹…βˆ‡)π‘ž+π‘„π‘žπ‘‘π‘₯.(4.5) Then, 12π‘‘ξ€œπ‘‘π‘‘Ξ©π‘ž2ξ€œπ‘‘π‘₯=Ξ©ξ‚€βˆ’πΏπ‘’||||βˆ‡π‘ž2+𝑒2ξ‚π‘žπ‘‘π‘₯+πΊπ‘žπ‘‘π‘₯.(4.6)
We deduce from (4.2)–(4.6) the following: 12π‘‘ξ€œπ‘‘π‘‘Ξ©ξ‚΅1π‘ƒπ‘Ÿπ‘’2+𝑇2+π‘ž2ξ‚Άξ€œπ‘‘π‘₯=Ξ©ξ‚ƒβˆ’||||βˆ‡π‘’2βˆ’||||βˆ‡π‘‡2βˆ’πΏπ‘’||||βˆ‡π‘ž2βˆ’πœŽπ‘’β‹…π‘’+(𝑅+1)𝑇𝑒2βˆ’ξ‚€ξ‚ξ‚π‘…βˆ’1π‘žπ‘’2ξ‚„β‰€ξ€œ+𝑄𝑇+πΊπ‘žπ‘‘π‘₯Ξ©ξ‚ƒβˆ’||||βˆ‡π‘’2βˆ’||||βˆ‡π‘‡2βˆ’πΏπ‘’||||βˆ‡π‘ž2βˆ’ξ‚πœŽ|𝑒|2||𝑒+ξ‚πœŽ2||2+(𝑅+1)2||𝑇||2ξ‚πœŽ2+ξ‚€ξ‚ξ‚π‘…βˆ’12||π‘ž||2ξ‚πœŽ2||𝑇||+πœ€2||π‘ž||+πœ€2+1πœ€ξ‚€||𝑄||2+||𝐺||2𝑑π‘₯.(4.7) Let πœ€>0 be appropriate small such that π‘‘ξ€œπ‘‘π‘‘Ξ©ξ€·π‘’2+𝑇2+π‘ž2𝑑π‘₯≀𝐢1ξ€œΞ©ξ‚ƒβˆ’||||βˆ‡π‘’2βˆ’||||βˆ‡π‘‡2βˆ’||||βˆ‡π‘ž2𝑑π‘₯+𝐢2ξ€œΞ©ξ‚€||𝑄||2+||𝐺||2𝑑π‘₯.(4.8)
Then, π‘‘ξ€œπ‘‘π‘‘Ξ©ξ€·π‘’2+𝑇2+π‘ž2𝑑π‘₯β‰€βˆ’πΆ3ξ€œΞ©ξ‚€|𝑒|2+||𝑇||2+||π‘ž||2𝑑π‘₯+𝐢4.(4.9)
Applying the Gronwall inequality, it follows that β€–β€–(𝑒,𝑇,π‘ž)(𝑑)2𝐿2≀‖(𝑒,𝑇,π‘ž)(0)β€–2𝐿2π‘’βˆ’πΆ3𝑑+𝐢4𝐢3ξ€·1βˆ’π‘’βˆ’πΆ3𝑑.(4.10)
Then, when 𝑀2>𝐢4/𝐢3, for any (𝑒0,𝑇0,π‘ž0)∈𝐡, here 𝐡 is a bounded in 𝐻, there exists π‘‘βˆ—>0 such that 𝑆𝑒(𝑑)0,𝑇0,π‘ž0ξ€Έ=(𝑒(𝑑),𝑇(𝑑),π‘ž(𝑑))βˆˆπ΅π‘€,𝑑>π‘‘βˆ—,(4.11) where 𝐡𝑀 is a ball in 𝐻, at 0 of radius 𝑀. Thus, (1.1)–(1.7) have an absorbing 𝐡𝑀 in 𝐻.
Step  2. 𝐢-condition is satisfied.
The eigenvalue equation: 𝑒π‘₯Δ𝑒=πœ†π‘’,1ξ€Έξ€·π‘₯,0=𝑒2𝑒,0=0,0,π‘₯2ξ€Έξ€·=𝑒2πœ‹,π‘₯2ξ€Έ,div𝑒=0(4.12) has eigenvalues πœ†1,πœ†2,…,πœ†π‘˜,… and eigenvector {π‘’π‘˜βˆ£π‘˜=1,2,3,…}, and πœ†1β‰₯πœ†2β‰₯β‹―β‰₯πœ†π‘˜β‰₯β‹―. If π‘˜β†’βˆž, then πœ†π‘˜β†’βˆ’βˆž. {π‘’π‘˜βˆ£π‘˜=1,2,3,…} constitutes an orthogonal base of 𝐿2(Ξ©).
For all (𝑒,𝑇,π‘ž)∈𝐻, we have 𝑒=βˆžξ“π‘˜=1π‘’π‘˜π‘’π‘˜,‖𝑒‖2𝐿2=βˆžξ“π‘˜=1𝑒2π‘˜,𝑇=βˆžξ“π‘˜=1π‘‡π‘˜π‘’π‘˜,‖𝑇‖2𝐿2=βˆžξ“π‘˜=1𝑇2π‘˜,π‘ž=βˆžξ“π‘˜=1π‘žπ‘˜π‘’π‘˜,β€–π‘žβ€–2𝐿2=βˆžξ“π‘˜=1π‘žπ‘˜.(4.13)
When π‘˜β†’βˆž, πœ†π‘˜β†’βˆ’βˆž. Let 𝛿 be small positive constant, and 𝑁=1/𝛿. There exists positive integer π‘˜ such that βˆ’π‘β‰₯πœ†π‘—,𝑗β‰₯π‘˜+1.(4.14)
Introduce subspace 𝐸1=span{𝑒1,𝑒2,…,π‘’π‘˜}βŠ‚πΏ2(Ξ©). Let 𝐸2 be an orthogonal subspace of 𝐸1 in 𝐿2(Ξ©).
For all (𝑒,𝑇,π‘ž)∈𝐻, we find that 𝑒=𝑣1+𝑣2,𝑇=𝑇1+𝑇2,π‘ž=π‘ž1+π‘ž2,𝑣1=π‘˜ξ“π‘–=1π‘₯π‘–π‘’π‘–βˆˆπΈ1×𝐸1,𝑣2=βˆžξ“π‘—=π‘˜+1π‘₯π‘—π‘’π‘—βˆˆπΈ2×𝐸2,𝑇1=π‘˜ξ“π‘–=1π‘‡π‘–π‘’π‘–βˆˆπΈ1,𝑇2=βˆžξ“π‘—=π‘˜+1π‘‡π‘—π‘’π‘—βˆˆπΈ2,π‘ž1=π‘˜ξ“π‘–=1π‘žπ‘–π‘’π‘–βˆˆπΈ1,π‘ž2=βˆžξ“π‘—=π‘˜+1π‘žπ‘—π‘’π‘—βˆˆπΈ2.(4.15)
Let π‘ƒπ‘–βˆΆπΏ2(Ξ©)→𝐸𝑖 be the orthogonal projection. Thanks to Definition 2.3, we will prove that for any bounded set π΅βŠ‚π» and πœ€>0, there exists 𝑑0>0 such that ‖‖𝑃1‖‖𝑆(𝑑)𝐡𝐻≀𝑀,βˆ€π‘‘>𝑑0‖‖𝑃,𝑀isaconstant,(4.16)2‖‖𝑆(𝑑)π΅π»β‰€πœ€,βˆ€π‘‘>𝑑0,𝑒0,𝑇0,π‘ž0ξ€Έβˆˆπ΅.(4.17)
From Step  1, 𝑆(𝑑) has an absorbing set 𝐡𝑀. Then for any bounded set π΅βŠ‚π», there exists π‘‘βˆ—>0 such that 𝑆(𝑑)π΅βŠ‚π΅π‘€, for all 𝑑>π‘‘βˆ—, which imply (4.16).
Multiply (1.1) by 𝑒 and integrate over (Ξ©). We obtain 𝑑𝑒𝑑𝑑,𝑒=π‘ƒπ‘Ÿ(Δ𝑒,𝑒)βˆ’π‘ƒπ‘Ÿ(πœŽπ‘’,𝑒)+π‘ƒπ‘Ÿξ‚ξ‚ξ‚€ξ‚€π‘…π‘‡+π‘…π‘žβ†’ξ‚πœ…,π‘’βˆ’((π‘’β‹…βˆ‡)𝑒,𝑒).(4.18) Then, ‖𝑒‖2𝐿2=π‘ƒπ‘Ÿξ€œπ‘‘0(Δ𝑒,𝑒)π‘‘π‘‘βˆ’π‘ƒπ‘Ÿξ€œπ‘‘0(πœŽπ‘’,𝑒)𝑑𝑑+π‘ƒπ‘Ÿξ€œπ‘‘0𝑅𝑇+π‘…π‘žβ†’ξ‚β€–β€–π‘’πœ…,𝑒𝑑𝑑+0β€–β€–2𝐿2=πœ€1π‘ƒπ‘Ÿξ€œπ‘‘0ξ€·(Δ𝑒,𝑒)𝑑𝑑+1βˆ’πœ€1ξ€Έπ‘ƒπ‘Ÿξ€œπ‘‘0(Δ𝑒,𝑒)π‘‘π‘‘βˆ’π‘ƒπ‘Ÿξ€œπ‘‘0(πœŽπ‘’,𝑒)𝑑𝑑+π‘ƒπ‘Ÿξ€œπ‘‘0𝑅𝑇+π‘…π‘žβ†’ξ‚β€–β€–π‘’πœ…,𝑒𝑑𝑑+0β€–β€–2𝐿2,(4.19) where πœ€1 is a constant which needs to be determined.
From (4.14), we find that (Δ𝑒,𝑒)=βˆžξ“π‘–=1πœ†π‘–π‘’2𝑖=π‘˜ξ“π‘–=1πœ†π‘–π‘’2𝑖+βˆžξ“π‘—=π‘˜+1πœ†π‘—π‘’2π‘—β‰€πœ†π‘˜ξ“π‘–=1𝑒2π‘–βˆ’π‘βˆžξ“π‘–=π‘˜+1𝑒2π‘—β‰€πœ†β€–π‘’β€–2𝐿2β€–β€–πœˆβˆ’π‘2β€–β€–2𝐿2,(4.20) where πœ†=max{πœ†1,πœ†2,…,πœ†π‘˜}.
Thanks to ∫(Δ𝑒,𝑒)=βˆ’Ξ©|βˆ‡π‘’|2𝑑π‘₯=βˆ’β€–βˆ‡π‘’β€–2𝐿2 and ‖𝑒‖𝐿2β‰€πΆβ€–βˆ‡π‘’β€–πΏ2, it follows that (Δ𝑒,𝑒)=βˆ’β€–βˆ‡π‘’β€–2𝐿21β‰€βˆ’πΆ2‖𝑒‖2𝐿2.(4.21)
We deduce from (4.11) the following: ‖𝑒‖2𝐿2+‖𝑇‖2𝐿2+β€–π‘žβ€–2𝐿2≀𝑀2,𝑑β‰₯π‘‘βˆ—.(4.22)
Using (4.19)–(4.22), we find that β€–β€–πœˆ2β€–β€–2𝐿2≀‖𝑒‖2𝐿2=πœ€1π‘ƒπ‘Ÿξ€œπ‘‘0ξ€·(Δ𝑒,𝑒)𝑑𝑑+1βˆ’πœ€1ξ€Έπ‘ƒπ‘Ÿξ€œπ‘‘0(Δ𝑒,𝑒)π‘‘π‘‘βˆ’π‘ƒπ‘Ÿξ€œπ‘‘0(πœŽπ‘’,𝑒)𝑑𝑑+π‘ƒπ‘Ÿξ€œπ‘‘0𝑅𝑇+π‘…π‘žβ†’ξ‚β€–β€–π‘’πœ…,𝑒𝑑𝑑+0β€–β€–2𝐿2β‰€πœ€1πœ†π‘ƒπ‘Ÿξ€œπ‘‘0‖𝑒‖2𝐿2π‘‘π‘‘βˆ’πœ€1π‘π‘ƒπ‘Ÿξ€œπ‘‘0β€–β€–πœˆ2β€–β€–2𝐿2ξ€·π‘‘π‘‘βˆ’1βˆ’πœ€1ξ€Έπ‘ƒπ‘ŸπΆ2ξ€œπ‘‘0‖𝑒‖2𝐿2+π‘ƒπ‘‘π‘‘π‘Ÿπ‘…2ξ‚΅ξ€œπ‘‘0‖𝑒‖2𝐿2ξ€œπ‘‘π‘‘+𝑑0‖𝑇‖2𝐿2ξ‚Ά+π‘ƒπ‘‘π‘‘π‘Ÿξ‚π‘…2ξ‚΅ξ€œπ‘‘0‖𝑒‖2𝐿2ξ€œπ‘‘π‘‘+𝑑0β€–π‘žβ€–2𝐿2ξ‚Ά+‖‖𝑒𝑑𝑑0β€–β€–2𝐿2.(4.23) Let πœ€1 satisfy πœ€1πœ†β‰€(1βˆ’πœ€1)/𝐢2 and 𝐾1=π‘ƒπ‘Ÿπ‘…π‘€+π‘ƒπ‘Ÿξ‚π‘…π‘€. Then, β€–β€–πœˆ2β€–β€–2𝐿2β‰€βˆ’πœ€1π‘π‘ƒπ‘Ÿξ€œπ‘‘0β€–β€–πœˆ2β€–β€–2𝐿2𝑑𝑑+𝐾1‖‖𝑒𝑑+0β€–β€–2𝐿2,𝑑>π‘‘βˆ—.(4.24) By the Gronwall inequality, we find that β€–β€–πœˆ2β€–β€–2𝐿2β‰€π‘’βˆ’πœ€1π‘π‘ƒπ‘Ÿπ‘‘β€–β€–π‘’0β€–β€–2𝐿2+𝐾1πœ€1π‘π‘ƒπ‘Ÿξ€·1βˆ’π‘’βˆ’πœ€1π‘π‘ƒπ‘Ÿπ‘‘ξ€Έ,𝑑>π‘‘βˆ—.(4.25) Then, there exists 𝑑1>π‘‘βˆ— satisfying π‘’βˆ’πœ€1π‘π‘ƒπ‘Ÿπ‘‘1‖‖𝑒0β€–β€–2𝐿2≀𝐾12πœ€1π‘π‘ƒπ‘Ÿ,𝐾1πœ€1π‘π‘ƒπ‘Ÿξ€·1βˆ’π‘’βˆ’πœ€1π‘π‘ƒπ‘Ÿπ‘‘1≀𝐾12πœ€1π‘π‘ƒπ‘Ÿ.(4.26) Since 𝛿=1/𝑁, for 𝑑>𝑑1 it follows that β€–β€–πœˆ2β€–β€–2𝐿2≀𝐾1πœ€1π‘π‘ƒπ‘Ÿ=𝐾1πœ€1π‘ƒπ‘Ÿπ›Ώ.(4.27)
Multiply (1.2) by 𝑇 and integrate over (Ξ©). We obtain 𝑑𝑇𝑒𝑑𝑑,𝑇=(Δ𝑇,𝑇)+2ξ€Έ,π‘‡βˆ’((π‘’β‹…βˆ‡)𝑇,𝑇)+(𝑄,𝑇).(4.28) Then, ‖𝑇‖2𝐿2=ξ€œπ‘‘0(ξ€œΞ”π‘‡,𝑇)𝑑𝑑+𝑑0𝑒2‖‖𝑇,𝑇𝑑𝑑+0β€–β€–2𝐿2=πœ€2ξ€œπ‘‘0ξ€·(Δ𝑇,𝑇)𝑑𝑑+1βˆ’πœ€2ξ€Έξ€œπ‘‘0ξ€œ(Δ𝑇,𝑇)𝑑𝑑+𝑑0𝑒2‖‖𝑇,𝑇𝑑𝑑+0β€–β€–2𝐿2,(4.29) where πœ€2 is a constant which needs to be determined.
From (4.14), we find that (Δ𝑇,𝑇)=βˆžξ“π‘–=1πœ†π‘–π‘‡2𝑖=π‘˜ξ“π‘–=1πœ†π‘–π‘‡2𝑖+βˆžξ“π‘—=π‘˜+1πœ†π‘—π‘‡2π‘—β‰€πœ†π‘˜ξ“π‘–=1𝑇2π‘–βˆ’π‘βˆžξ“π‘—=π‘˜+1𝑇2π‘—β‰€πœ†β€–π‘‡β€–2𝐿2β€–β€–π‘‡βˆ’π‘2β€–β€–2𝐿2,(4.30) where πœ†=max{πœ†1,πœ†2,β‹―,πœ†π‘˜}.
Since ∫(Δ𝑇,𝑇)=βˆ’Ξ©|βˆ‡π‘‡|2𝑑π‘₯=βˆ’β€–βˆ‡π‘‡β€–2𝐿2 and ‖𝑇‖𝐿2β‰€πΆβ€–βˆ‡π‘‡β€–πΏ2, it follows that (Δ𝑇,𝑇)=βˆ’β€–βˆ‡π‘‡β€–2𝐿21β‰€βˆ’πΆ2‖𝑇‖2𝐿2.(4.31)
Using (4.22) and (4.29)–(4.31), we find that ‖‖𝑇2β€–β€–2𝐿2≀‖𝑇‖2𝐿2=πœ€2ξ€œπ‘‘0ξ€·(Δ𝑇,𝑇)𝑑𝑑+1βˆ’πœ€2ξ€Έξ€œπ‘‘0ξ€œ(Δ𝑇,𝑇)𝑑𝑑+𝑑0𝑒2‖‖𝑇,𝑇𝑑𝑑+0β€–β€–2𝐿2β‰€πœ€2πœ†ξ€œπ‘‘0‖𝑇‖2𝐿2π‘‘π‘‘βˆ’πœ€2π‘ξ€œπ‘‘0‖‖𝑇2β€–β€–2𝐿2ξ€·π‘‘π‘‘βˆ’1βˆ’πœ€2𝐢2ξ€œπ‘‘0‖𝑇‖2𝐿2+1𝑑𝑑2ξ‚΅ξ€œπ‘‘0‖𝑒‖2𝐿2ξ€œπ‘‘π‘‘+𝑑0‖𝑇‖2𝐿2ξ‚Ά+‖‖𝑇𝑑𝑑0β€–β€–2𝐿2.(4.32) Let πœ€2 satisfy πœ€2πœ†β‰€(1βˆ’πœ€2)/𝐢2. Then ‖‖𝑇2β€–β€–2𝐿2β‰€βˆ’πœ€2π‘ξ€œπ‘‘0‖‖𝑇2β€–β€–2𝐿2‖‖𝑇𝑑𝑑+𝑀𝑑+0β€–β€–2𝐿2,𝑑>π‘‘βˆ—.(4.33) By the Gronwall inequality, we find that ‖‖𝑇2β€–β€–2𝐿2β‰€π‘’βˆ’πœ€2𝑁𝑑‖‖𝑇0β€–β€–2𝐿2+π‘€πœ€2𝑁1βˆ’π‘’βˆ’πœ€2𝑁𝑑,𝑑>π‘‘βˆ—.(4.34) Then, there exists 𝑑2>π‘‘βˆ— satisfying π‘’βˆ’πœ€2𝑁𝑑2‖‖𝑇0β€–β€–2𝐿2≀𝑀2πœ€2𝑁,π‘€πœ€2𝑁1βˆ’π‘’βˆ’πœ€2𝑁𝑑2≀𝑀2πœ€2𝑁.(4.35) Since 𝛿=1/𝑁, for 𝑑>𝑑2, it follows that ‖‖𝑇2β€–β€–2𝐿2β‰€π‘€πœ€2𝑁=π‘€πœ€2𝛿.(4.36)
Multiply (1.3) by π‘ž and integrate over (Ξ©). We obtain ξ‚΅π‘‘π‘žξ‚Άπ‘‘π‘‘,π‘ž=𝐿𝑒𝑒(Ξ”π‘ž,π‘ž)+2ξ€Έ,π‘žβˆ’((π‘’β‹…βˆ‡)π‘ž,π‘ž)+(𝐺,π‘ž).(4.37) Then, β€–π‘žβ€–2𝐿2=πΏπ‘’ξ€œπ‘‘0(ξ€œΞ”π‘ž,π‘ž)𝑑𝑑+𝑑0𝑒2ξ€Έβ€–β€–π‘ž,π‘žπ‘‘π‘‘+0β€–β€–2𝐿2=πΏπ‘’πœ€3ξ€œπ‘‘0ξ€·(Ξ”π‘ž,π‘ž)𝑑𝑑+1βˆ’πœ€3ξ€Έξ€œπ‘‘0ξ€œ(Ξ”π‘ž,π‘ž)𝑑𝑑+𝑑0𝑒2ξ€Έβ€–β€–π‘ž,π‘žπ‘‘π‘‘+0β€–β€–2𝐿2,(4.38) where πœ€3 is a constant which needs to be determined.
From (4.14), we find that (Ξ”π‘ž,π‘ž)=βˆžξ“π‘–=1πœ†π‘–π‘ž2𝑖=π‘˜ξ“π‘–=1πœ†π‘–π‘ž2𝑖+βˆžξ“π‘—=π‘˜+1πœ†π‘—π‘ž2π‘—β‰€πœ†π‘˜ξ“π‘–=1π‘ž2π‘–βˆ’π‘βˆžξ“π‘—=π‘˜+1π‘ž2π‘—β‰€πœ†β€–π‘žβ€–2𝐿2β€–β€–π‘žβˆ’π‘2β€–β€–2𝐿2,(4.39) where πœ†=max{πœ†1,πœ†2,…,πœ†π‘˜}.
Since ∫(Ξ”π‘ž,π‘ž)=βˆ’Ξ©|βˆ‡π‘ž|2𝑑π‘₯=βˆ’β€–βˆ‡π‘žβ€–2𝐿2 and β€–π‘žβ€–πΏ2β‰€πΆβ€–βˆ‡π‘žβ€–πΏ2, we see that (Ξ”π‘ž,π‘ž)=βˆ’β€–βˆ‡π‘žβ€–2𝐿21β‰€βˆ’πΆ2β€–π‘žβ€–2𝐿2.(4.40)
Using (4.22) and (4.38)–(4.40), we obtain β€–β€–π‘ž2β€–β€–2𝐿2β‰€β€–π‘žβ€–2𝐿2=πœ€3πΏπ‘’ξ€œπ‘‘0ξ€·(Ξ”π‘ž,π‘ž)𝑑𝑑+1βˆ’πœ€3ξ€ΈπΏπ‘’ξ€œπ‘‘0ξ€œ(Ξ”π‘ž,π‘ž)𝑑𝑑+𝑑0𝑒2ξ€Έβ€–β€–π‘ž,π‘žπ‘‘π‘‘+0β€–β€–2𝐿2β‰€πœ€3πœ†πΏπ‘’ξ€œπ‘‘0β€–π‘žβ€–2𝐿2π‘‘π‘‘βˆ’πœ€3π‘πΏπ‘’ξ€œπ‘‘0β€–β€–π‘ž2β€–β€–2𝐿2π‘‘π‘‘βˆ’1βˆ’πœ€3𝐢2πΏπ‘’ξ€œπ‘‘0β€–π‘žβ€–2𝐿2+1𝑑𝑑2ξ‚΅ξ€œπ‘‘0‖𝑒‖2𝐿2ξ€œπ‘‘π‘‘+𝑑0β€–π‘žβ€–2𝐿2ξ‚Ά+β€–β€–π‘žπ‘‘π‘‘0β€–β€–2𝐿2.(4.41) Let πœ€3 satisfy πœ€3πœ†β‰€(1βˆ’πœ€3)/𝐢2. Then, β€–β€–π‘ž2β€–β€–2𝐿2β‰€βˆ’πœ€3π‘πΏπ‘’ξ€œπ‘‘0β€–β€–π‘ž2β€–β€–2𝐿2β€–β€–π‘žπ‘‘π‘‘+𝑀𝑑+0β€–β€–2𝐿2,𝑑>π‘‘βˆ—.(4.42) By the Gronwall inequality, we find that β€–β€–π‘ž2β€–β€–2𝐿2β‰€π‘’βˆ’πœ€3πΏπ‘’π‘π‘‘β€–β€–π‘ž0β€–β€–2𝐿2+π‘€πœ€3𝐿𝑒𝑁1βˆ’π‘’βˆ’πœ€3𝐿𝑒𝑁𝑑,𝑑>π‘‘βˆ—.(4.43) Then, there exists 𝑑3>π‘‘βˆ— satisfying π‘’βˆ’πœ€3𝐿𝑒𝑁𝑑3β€–β€–π‘ž0β€–β€–2𝐿2≀𝑀2πœ€3𝐿𝑒𝑁,π‘€πœ€3𝐿𝑒𝑁1βˆ’π‘’βˆ’πœ€3𝐿𝑒𝑁𝑑3≀𝑀2πœ€3𝐿𝑒𝑁.(4.44) Since 𝛿=1/𝑁, for 𝑑>𝑑3, it follows that β€–β€–π‘ž2β€–β€–2𝐿2β‰€π‘€πœ€3𝐿𝑒𝑁=π‘€πœ€3𝐿𝑒𝛿.(4.45)
From (4.27); (4.36) and (4.45) for all 𝛿>0 there exists 𝑑0=max{𝑑1,𝑑2,𝑑3} such that when 𝑑>𝑑0, it follows that ‖‖𝑃2𝑒𝑆(𝑑)0,𝑇0,π‘ž0ξ€Έβ€–β€–2𝐻=‖‖𝑣2β€–β€–2𝐿2+‖‖𝑇2β€–β€–2𝐿2+β€–β€–π‘ž2β€–β€–2𝐿2≀𝐾1πœ€1π‘ƒπ‘Ÿ+π‘€πœ€2+π‘€πœ€3𝐿𝑒𝛿,(4.46) which imply (4.17). From Lemma 2.4, (1.1)–(1.7) have a global attractor in 𝐿2(Ξ©,𝑅4).


This work was supported by national natural science foundation of China (no. 11271271) and the NSF of Sichuan Education Department of China (no. 11ZA102).


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