Strong Global Attractors for 3D Wave Equations with Weakly Damping

Abstract

We consider the existence of the global attractor 𝒜₁ for the 3D weakly damped wave equation. We prove that 𝒜₁ is compact in (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) and attracts all bounded subsets of (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) with respect to the norm of (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω). Furthermore, this attractor coincides with the global attractor in the weak energy space 𝐻¹₀(Ω)×𝐿²(Ω).

1. Introduction

Let Ω⊂𝑅³ be a bounded domain with smooth boundary 𝜕Ω. We consider the following weakly damped wave equation: 𝑢_𝑡𝑡+𝛼𝑢_𝑡−Δ𝑢+𝜑(𝑢)=𝑓inΩ×𝑅⁺ (1.1) with the boundary condition𝑢∣_𝜕Ω=0, (1.2) and initial conditions:𝑢(⋅,0)=𝑢₀,𝑢_𝑡(⋅,0)=𝑢₁,inΩ, (1.3) where 𝛼>0, 𝜑 is the nonlinear term, and 𝑓 is a given external forcing term.

Nonlinear wave equation of the type (1.1) arises as an evolutionary mathematical model in many branched of physics, for example, (i) modeling a continuous Josephson junction with 𝜑(𝑢)=𝛽sin𝑢; (ii) modeling a relativistic quantum mechanics with 𝜑(𝑢)=|𝑢|^𝛾𝑢. A relevant problem is to investigate the asymptotic dynamical behavior of these mathematical models. The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem is to analyse the existence of its global attractor.

The existence of global attractors for the classical wave equations in 𝐻¹₀(Ω)×𝐿²(Ω) and the regularities of the global attractors has been studied extensively in many monographs and lectures, for example, see [1–7] and references therein.

However, to our knowledge, the research about the stronger attraction of global attractors for the damped wave equations with respect to the norm of (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) is fewer, only has been found in [8–10]. In the above three papers, the global attractors in strong topological space (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) were established, the attraction with respect to the norm of (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) was proved by the asymptotic compactness of the operator semigroup.

Recently, we consider (1.1) in 𝑛 dimensional space where the nonlinear term 𝜑 without polynomial growth is in [11].

In this paper, our aim is to prove the existence of a global attractor for (1.1) in strong topological space (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) where the nonlinear term 𝜑 with some polynomial growth. For simplicity, we consider the space dimension is 3, as we know, when the space dimension is lagerer than 3, the case is similar as in 3D, when the space dimension is 1 or 2, the case is more easier. The attraction with respect to the norm of (𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω) will be proved by a method different from [8–10]. Furthermore, this attractor coincides with the global attractor in the weak energy space 𝐻¹₀(Ω)×𝐿²(Ω).

The basic assumptions about the external forcing term 𝑓 and the nonlinear term 𝜑 are as follows. Let 𝑓∈𝐿²(Ω) be independent of time, and let the nonlinear term 𝜑∈𝐶¹(𝑅,𝑅) satisfy the following assumptions:liminf_|𝑠|→∞Φ(𝑠)𝑠²≥0,hereΦ(𝑠)=^𝑠₀𝜑(𝜏)𝑑𝜏;limsup_|𝑠|→∞𝜑^′(𝑠)𝑠²=0; (1.4)(1.5) moreover, there exists a constant 𝐶₀>0 such thatliminf_|𝑠|→∞𝑠𝜑(𝑠)−𝐶₀Φ(𝑠)𝑠²≥0. (1.6)

Throughout this paper, we use the following notations. Let Ω be a bounded subset of 𝑅^𝑛 with sufficiently smooth boundary, 𝐴=−Δ. 𝑉=𝐻¹₀(Ω), 𝐻=𝐿²(Ω), and 𝐷(𝐴)=𝐻²(Ω)∩𝐻¹₀(Ω) with the corresponding norms ‖𝑢‖=(∫_Ω|∇𝑢(𝑥)|²𝑑𝑥)^1/2, |𝑢|=(∫_Ω|𝑢(𝑥)|²𝑑𝑥)^1/2 and |Δ𝑢|=(∫_Ω|Δ𝑢(𝑥)|²𝑑𝑥)^1/2, respectively. The norms in 𝐿^𝑝(Ω),1≤𝑝<∞ are denoted by |𝑢|_𝑝=(∫_Ω|𝑢|^𝑝𝑑𝑥)^1/𝑝, the scalar products of 𝑉,𝐻 are denoted by((𝑢,𝑣))=_Ω∇𝑢(𝑥)∇𝑣(𝑥)𝑑𝑥,(𝑢,𝑣)=_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥, (1.7) respectively. We have 𝐷(𝐴)⊂𝑉⊂𝐻=𝐻^∗⊂𝑉^∗,𝐻^∗ and 𝑉^∗ are the dual spaces of 𝐻 and 𝑉, respectively, and each space is dense in the following one and the injections are continuous. Then, we introduce the product Hilbert spaces ℋ₀=𝑉×𝐻=𝐻¹₀(Ω)×𝐿²(Ω),ℋ₁=𝐷(𝐴)×𝑉=(𝐻²(Ω)∩𝐻¹₀(Ω))×𝐻¹₀(Ω), endowed with the standard product norms:𝑢₁,𝑢₂²_ℋ0=𝑢₁²_𝑉+𝑢₂²_𝐻,𝑢₁,𝑢₂²_ℋ1=𝑢₁²_𝐷(𝐴)+𝑢₂²_𝑉. (1.8) Denote by 𝐶 any positive constant which may be different from line to line and even in the same line, we also denote the different positive constants by 𝐶_𝑖,𝑖∈ℕ, for special differentiation.

The rest of the paper is organized as follows. In the next section, for the convenience of the reader, we recall some basic concepts about the global attractors and recapitulate some abstract results. In Section 3, we present our main results.

2. Preliminaries

In this section, we first recall some basic concepts and theorems, which are important for getting our main results. We refer to [2, 5, 6, 12, 13] and the references therein for more details. Then, we outline some known results about (1.1)–(1.3).

Definition 2.1. The mappings 𝑆(𝑡), where 𝑆∶𝑋×[0,+∞)→𝑋, is said to be a 𝐶⁰ semigroup on 𝑋, if {𝑆(𝑡)}_𝑡≥0 satisfies(1)𝑆(0)𝑢=𝑢 for all 𝑢∈𝑋;(2)𝑆(𝑡₁)(𝑆(𝑡₂)𝑢)=𝑆(𝑡₁+𝑆₂)(𝑢) for all 𝑢∈𝑋 and 𝑡₁,𝑡₂∈𝑅⁺;(3)the mapping 𝑆∶𝑋×(0,∞)→𝑋 is continuous.

Definition 2.2. Let {𝑆(𝑡)}_𝑡≥0 be a semigroup on a metric space (𝐸,𝑑). A subset 𝒜 of 𝐸 is called a global attractor for the semigroup, if 𝒜 is compact and enjoys the following properties:(1)𝒜 is invariant, that is, 𝑆(𝑡)𝒜=𝒜, forall𝑡≥0;(2)𝒜 attracts all bounded sets of 𝐸. That is, for any bounded subset 𝐵 of 𝐸,𝑑(𝑆(𝑡)𝐵,𝒜)⟶0,as𝑡⟶0, (2.1) where 𝑑(𝐵,𝐴) is the semidistance of two sets B and A: 𝑑(𝐵,𝐴)=sup_𝑥∈𝐵inf_𝑦∈𝐴𝑑𝑥,𝑦. (2.2)

Definition 2.3. A 𝐶⁰ semigroup {𝑆(𝑡)}_𝑡≥0 in a Banach space 𝑋 is said to satisfy the condition (𝐶) if for any 𝜀>0 and for any bounded set 𝐵 of 𝑋, there exist 𝑡(𝐵)>0 and a finite dimensional subspace 𝑋₁ of 𝑋 such that {‖𝑃𝑆(𝑡)𝑥‖_𝑋,𝑥∈𝐵,𝑡≥𝑡(𝐵)} is bounded and ‖(𝐼−𝑃)𝑆(𝑡)𝑥‖_𝑋<𝜀,𝑡≥𝑡(𝐵),𝑥∈𝐵, (2.3) where 𝑃∶𝑋→𝑋₁ is a bounded projector.

Definition 2.4. Let {𝑆(𝑡)}_𝑡≥0 be a semigroup on a metric space (𝐸,𝑑). A set 𝐵₀⊂𝐸 is called an absorbing set for the semigroup {𝑆(𝑡)}_𝑡≥0, if and only if for every bounded set 𝐵⊂𝐸, there exists a 𝑇₀=𝑇₀(𝐵)>0 such that 𝑆(𝑡)𝐵⊂𝐵₀ for all 𝑡≥𝑇₀.

Theorem 2.5. Let 𝑋 be a Banach space and let {𝑆(𝑡)}_𝑡≥0 be a 𝐶⁰ semigroup in 𝑋. Then, there is a global attractor for {𝑆(𝑡)}_𝑡≥0 in 𝑋 if the following conditions hold true:(1){𝑆(𝑡)}_𝑡≥0 satisfies the condition (𝐶), and(2)there is a bounded absorbing set 𝐵⊂𝑋.

In [12], the authors have discussed the relations between Condition (𝐶) and 𝜔-limit compact and proved that, in uniformly convex Banach space, Condition (𝐶) is equivalent to 𝜔-limit compact, if the semigroup has a bounded absorbing set.

Next, we recall the result about the global attractor in ℋ₀ whose proofs are omitted here, the reader is referred to [6] and the reference therein.

Theorem 2.6. Under the conditions (1.4), (1.5), (1.6), the solution semigroup {𝑆(𝑡)}_𝑡≥0 of the problem (1.1)–(1.3) has a global attractor 𝒜₀ in ℋ₀. 𝒜₀ is included and bounded in ℋ₁.

3. Main Results

According to the standard Fatou-Galerkin method, it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value of (1.1)–(1.3). We address the reader to [6] and the reference therein. Here, we only state the result as follows.

Lemma 3.1. Let conditions (1.4), (1.5), (1.6) hold, then for any 𝑇>0 and (𝑢₀,𝑢₁)∈ℋ₀, there exists a unique solution of (1.1)–(1.3) such that 𝑢,𝑢_𝑡∈𝐶[0,𝑇];ℋ₀. (3.1) If, furthermore, 𝑢₀,𝑢₁∈ℋ₁, (3.2) then 𝑢 satisfies 𝑢,𝑢_𝑡∈𝐶[0,𝑇];ℋ₁. (3.3)

We define the mappings:𝑆(𝑡)∶𝑢₀,𝑢₁⟶𝑢(𝑡),𝑢_𝑡(𝑡)∀𝑡∈𝑅. (3.4) By Lemma 3.1, it is easy to see that {𝑆(𝑡)}_𝑡≥0 is 𝐶₀ semigroup in the energy phase spaces ℋ₀ and ℋ₁.

In order to verify the existence of the bounded absorbing set in ℋ₁, we need the result about the existence of the bounded absorbing set in ℋ₀. First, we establish the bounded absorbing set in ℋ₀. Its proof is essentially established in [6] and the reference therein, and we only need to make a few minor changes for our problem. Here, we only give the following lemma.

Lemma 3.2. Under the conditions (1.4), (1.5), (1.6), {𝑆(𝑡)}_𝑡≥0 has a bounded absorbing set ℬ₀≜𝐵_ℋ0(0,𝜌₀) in ℋ₀, that is, for any 𝜀>0 and any bounded subset 𝐵₀⊂ℋ₀, there is a positive constant 𝑡₀=𝑡(𝐵₀,𝜌₀) such that 𝑆(𝑡)𝐵⊂ℬ₀forany𝑡≥𝑡₀,𝑢₀,𝑢₁∈𝐵₀. (3.5)

Next, let us establish the existence of the bounded absorbing set in ℋ₁.

Lemma 3.3. Under the conditions (1.4), (1.5), (1.6), {𝑆(𝑡)}_𝑡≥0 has a bounded absorbing set ℬ≜𝐵_ℋ1(0,𝜌₁) in ℋ₁, that is, for any 𝜀>0 and any bounded subset 𝐵⊂ℋ₁, there is a positive constant 𝑇=𝑇(𝐵,𝜌₁) such that 𝑆(𝑡)𝐵⊂ℬforany𝑡≥𝑇,𝑢₀,𝑢₁∈𝐵. (3.6)

Proof. Take the scalar product in 𝐻 of (1.1) with 𝐴𝑣=𝐴𝑢_𝑡+𝜎𝐴𝑢, we have 12𝑑||𝑑𝑡𝐴𝑢²+‖𝑣‖²+𝜎|𝐴𝑢|²+(𝛼−𝜎)‖𝑣‖²−𝜎(𝛼−𝜎)(𝐴𝑢,𝑣)+𝜑(𝑢),𝐴𝑣=𝑓,𝐴𝑣. (3.7) For 0<𝜎≤𝜎₀, 𝜎₀={𝛼/4,𝜆₁/2𝛼}, by Hölder inequality, Poincaré inequality, and Cauchy inequality we have 𝜎|𝐴𝑢|²+(𝛼−𝜎)‖𝑣‖²()||−𝜎𝛼−𝜎)(𝐴𝑢,𝑣≥𝜎𝐴𝑢²+(𝛼−𝜎)‖𝑣‖²−𝜎(𝛼−𝜎)𝜆₁|𝑣‖||𝐴𝑢|‖≥𝜎𝐴𝑢²+34𝛼‖𝑣‖²−𝜎𝛼𝜆₁|𝑣‖||𝐴𝑢|‖≥𝜎𝐴𝑢²+34𝛼‖𝑣‖²−𝜎2|𝐴𝑢|²+𝜎𝛼²2𝜆₁‖𝑣‖²≥𝜎2|𝐴𝑢|²+𝛼2‖𝑣‖². (3.8) It follows from (1.5) that, for any 𝜀>0, there exists a constant 𝐶₁>0, such that 𝜑^′(𝑠)≤𝜀|𝑠|²+𝐶₁,∀𝑠∈ℝ. (3.9) Hence, 𝜑(𝑢)=,𝑣_Ω𝜑^′(𝑢)≤⋅∇𝑢⋅∇𝑣𝑑𝑥_Ω𝜑^′(𝑢)⋅|∇𝑢|⋅|∇𝑣|𝑑𝑥≤𝜀_Ω|𝑢|²⋅|∇𝑢|⋅|∇𝑣|𝑑𝑥+𝐶_1Ω|∇𝑢|⋅|∇𝑣|𝑑𝑥≤𝜀_Ω|𝑢|⁴⋅|∇𝑢|²𝑑𝑥^1/2⋅‖𝑣‖+𝐶₁‖𝑢‖⋅‖𝑣‖≤𝜀_Ω|𝑢|⁶𝑑𝑥^1/3⋅_Ω|∇𝑢|⁶𝑑𝑥^1/6⋅‖𝑣‖+𝐶₁‖𝑢‖⋅‖𝑣‖≤𝜀‖𝑢‖²⋅𝐶²₂⋅||⋅‖𝑣‖+𝛼𝐴𝑢‖𝑣‖16²+4𝐶²₁𝛼‖𝑢‖², (3.10) where 𝐶₂ is the positive constant satisfying 𝐶₂‖𝑢‖²≥_Ω|𝑢|⁶𝑑𝑥^1/3,𝐶₂|𝐴𝑢|≥_Ω|∇𝑢|⁶𝑑𝑥^1/6,=𝑑𝑓,𝐴𝑣≤𝑑𝑑𝑡𝑓,𝐴𝑢+𝜎𝑓,𝐴𝑢+𝜎𝑑𝑡𝑓,𝐴𝑢8|𝐴𝑢|²+2𝜎𝑓². (3.11)(3.12) If (𝑢₀,𝑣₀) belongs to a bounded set 𝐵 of ℋ₁, then 𝐵 is also bounded in ℋ₀, and for 𝑡≥𝑡₀, by Lemma 3.2, we have ‖𝑢‖²+𝑢_𝑡²≤𝜌²₀; (3.13)𝑡₀, 𝜌₀ are given in Lemma 3.2 Choose 0<𝜀²≤𝛼𝜎16𝜌⁴₀𝐶⁴₂, (3.14) it follows from (3.10) that 𝜑(𝑢)≤𝛼,𝑣8‖𝑣‖²+4𝜀²𝛼‖𝑢‖⁴𝐶⁴₂|𝐴𝑢|²+4𝐶²₁𝛼≤𝛼8‖𝑣‖²+𝜎4|𝐴𝑢|²+4𝐶²₁𝛼𝜌²₀,𝑡≥𝑡₀. (3.15) Combining with (3.8), (3.12), and (3.15), by the Hölder inequality and the Young inequality, we deduce from that (3.7): 𝑑||𝑑𝑡𝐴𝑢²+‖𝑣‖²+𝜎−2𝑓,𝐴𝑢4|𝐴𝑢|²+𝛼4|𝑣|²≤4𝜎𝑓²+8𝐶²₁𝛼𝜌²₀. (3.16) Let 𝑦=|𝐴𝑢−𝑓|²+‖𝑣‖², from the above inequality, we can obtain 𝑑𝑦+𝜎𝑑𝑡8𝑦≤𝑑𝑦+𝜎𝑑𝑡8|𝐴𝑢|²+‖𝑣‖²+𝑓²+𝜎4|𝑓=𝐴𝑢|𝑑𝑦+𝜎𝑑𝑡4|𝐴𝑢|²+‖𝑣‖²−𝜎8|𝐴𝑢|²+‖𝑣‖²+𝜎4|𝑓+𝜎𝐴𝑢|8𝑓²≤4𝜎𝑓²+8𝐶²₁𝛼𝜌²₀−𝜎8|𝐴𝑢|²+‖𝑣‖²+𝜎8|𝐴𝑢|²+𝑓²+𝜎8𝑓²≤4𝜎𝑓²+8𝐶²₁𝛼𝜌²₀+𝜎4𝑓²−𝜎8‖𝑣‖²≤𝜎4𝑓²+4𝜎𝑓²+8𝐶²₁𝛼𝜌²₀≤92𝜎𝑓²+8𝐶²₁𝛼𝜌²₀. (3.17) Let 𝐶₃=(9/2)𝜎|𝑓|²+(8𝐶²₁/𝛼)𝜌²₀. By the Gronwall lemma, we have 𝑦(𝑡)≤𝑦𝑡₀−𝜎exp8𝑡−𝑡₀+8𝐶₃𝜎. (3.18) Defining 𝜌^′₁ by 𝜌₁^′2=8𝐶₃𝜎, (3.19) we see that limsup_𝑡→∞𝑦(𝑡)≤𝜌₁^′2, (3.20) and we conclude that
The ball of ℋ₁, ℬ≜𝐵_ℋ1(0,𝜌₁), centered at (𝐴⁻¹𝑓,0) of radius 𝜌₁>𝜌^′₁+𝜌₀, is absorbing in ℋ₁ for the semigroup 𝑆(𝑡),𝑡≥0.

We now give the property of compactness about the nonlinear operator 𝜑 which will be needed in the proof of the condition (𝐶).

Lemma 3.4. Assume that 𝜑∈𝐶¹(𝑅,𝑅) and 𝜑∶𝐷(𝐴)→𝑉 are defined by 𝜑(𝑢),𝑣=_Ω𝜑^′(𝑢)∇𝑢∇𝑣𝑑𝑥, (3.21) for all 𝑢∈𝐷(𝐴),𝑣∈𝐻¹₀(Ω). Then, 𝜑 is continuous compact.

Proof. Let {𝑢_𝑚} be a bounded sequence in 𝐷(𝐴). Without loss of generality, we assume that {𝑢_𝑚} weakly converges to 𝑢₀ in 𝐷(𝐴), since 𝐷(𝐴) is reflexive. By the Sobolev embedding theorem, we know that 𝐻²(Ω)↪𝐿^∞⋂𝐻¹(Ω) and the embedding 𝐻²(Ω)↪𝐻¹(Ω) is compact in 𝑅³. Hence, we have that 𝑢_𝑚⟶𝑢₀in𝐻¹. (3.22) Furthermore, there exists a constant 𝐶 such that 𝑢_{0𝐿∞𝐻1(Ω)}≤𝐶,𝑢_{𝑚𝐿∞𝐻1(Ω)}≤𝐶. (3.23) It is sufficient to prove that {𝜑(𝑢_𝑚)} converges to {𝜑(𝑢₀)} in 𝑉: 𝜑𝑢_𝑚−𝜑𝑢₀=_Ω𝜑^′𝑢_𝑚∇𝑢_𝑚−𝜑^′𝑢₀∇𝑢₀²𝑑𝑥^1/2≤_Ω𝜑^′𝑢_𝑚²∇𝑢_𝑚−∇𝑢₀²𝑑𝑥^1/2+_Ω∇𝑢₀²𝜑^′𝑢_𝑚−𝜑^′𝑢₀²𝑑𝑥^1/2. (3.24) On the one hand, for the first term in (3.24), combining with (3.23) and the continuity of 𝜑′(⋅), we have _Ω𝜑^′𝑢_𝑚²∇𝑢_𝑚−∇𝑢₀²𝑑𝑥≤𝐶_{‖𝑢𝑚‖𝐿∞(Ω)}∇𝑢_𝑚−∇𝑢₀². (3.25) On the other hand, for the second term in (3.24), using the continuity of 𝜑′(⋅), _Ω∇𝑢₀²𝜑^′𝑢_𝑚−𝜑^′𝑢₀²𝑑𝑥⟶0,as𝑚⟶0, (3.26) follows immediately by dominated convergence theorem.
Also, considering (3.22), passing to the limit in (3.24), we can obtain lim_𝑚→∞𝜑𝑢_𝑚−𝜑𝑢₀=0. (3.27) This completes the proof.

Lemma 3.5. Suppose the conditions (1.4), (1.5), (1.6) hold, the solution semigroup {𝑆(𝑡)}_𝑡≥0 of the problem (1.1)–(1.3) satisfies the condition (𝐶) in ℋ₁.

Proof. Let {𝜔_𝑖} be an orthonormal basis of 𝐿²(Ω) which consists of eigenvalues of 𝐴. The corresponding eigenvalues are denoted by {𝜆_𝑗}^∞_𝑗=1: 0<𝜆₁≤𝜆₂≤𝜆₃≤⋯≤𝜆_𝑗≤⋯,𝜆_𝑗⟶∞ (3.28) with 𝐴𝜔_𝑖=𝜆_𝑖𝜔_𝑖,∀𝑖∈ℕ. (3.29) Let 𝑉_𝑚=  span{𝜔₁,…,𝜔_𝑚} in 𝑉 and let 𝑃_𝑚∶𝑉→𝑉_𝑚 be an orthogonal projector. We write 𝑢=𝑃_𝑚𝑢+𝐼−𝑃_𝑚𝑢≜𝑢₁+𝑢₂. (3.30) Taking the scalar product of (1.1) in 𝐻 with 𝐴𝑣₂=𝐴𝑢_2𝑡+𝜎𝐴𝑢₂, we find 12𝑑𝑑𝑡𝐴𝑢₂²+𝑣₂²+𝜎𝐴𝑢₂²+(𝛼−𝜎)𝑣₂²−𝜎(𝛼−𝜎)𝐴𝑢₂,𝑣₂+𝜑(𝑢),𝐴𝑣₂=𝑓,𝐴𝑣₂. (3.31) Choose 0<𝜎≤𝜎₀, similar to (3.8), we have 𝜎𝐴𝑢₂²+(𝛼−𝜎)𝑣₂²−𝜎(𝛼−𝜎)𝐴𝑢₂,𝑣₂≥𝜎2𝐴𝑢₂²+𝛼2𝑣₂². (3.32) Since 𝑓∈𝐿²(Ω), 𝜑∶𝐷(𝐴)→𝑉 is compact by Lemma 3.4, for any 𝜀>0, there exists some 𝑚 such that 𝐼−𝑃_𝑚𝑓_𝐻≤𝜀, (3.33)𝐼−𝑃_𝑚𝜑(𝑢)_𝑉≤𝜀,∀𝑢∈𝐵₁0,𝜌₁, (3.34) where 𝜌₁ is given by Lemma 3.2.
By exploiting the Hölder inequality and Cauchy inequality, we have 𝜑(𝑢),𝐴𝑣₂≤𝐼−𝑃_𝑚𝜑𝑣₂≤𝜀‖𝑣₂𝛼‖≤4𝑣₂²+𝜀²𝛼,=𝑑𝑓,𝐴𝑣≤𝑑𝑑𝑡𝑓,𝐴𝑢+𝜎𝑓,𝐴𝑢+𝜎𝑑𝑡𝑓,𝐴𝑢4|𝐴𝑢|²+𝜎𝑓². (3.35)(3.36) By (3.33) and (3.36), we have 𝑓,𝐴𝑣₂=𝑑𝑓𝑑𝑡²,𝐴𝑢₂+𝜎𝑓,𝐴𝑢₂≤𝑑𝑓𝑑𝑡²,𝐴𝑢₂+𝜎4𝐴𝑢₂²+𝜎𝑓₂²≤𝑑𝑓𝑑𝑡²,𝐴𝑢₂+𝜎4𝐴𝑢₂²+𝜎𝜀², (3.37) where 𝑓₂≜(𝐼−𝑃_𝑚)𝑓.
Hence, combining with (3.32), (3.35), and (3.37), we obtain from (3.31) that 𝑑𝑑𝑡𝐴𝑢₂²+𝑣₂²−2𝑓₂,𝐴𝑢₂+𝜎2𝐴𝑢₂²+𝑣₂²≤22𝜎+𝛼𝜀². (3.38) Let 𝑦=|𝐴𝑢₂−𝑓₂|²+‖𝑣₂‖², from the above inequality, similar to (3.17), we can obtain 𝑑𝑦+𝜎𝑑𝑡4𝑦≤𝑑𝑦+𝜎𝑑𝑡4𝐴𝑢₂²+𝑣₂²+𝑓₂²+𝜎2𝐴𝑢₂𝑓₂≤5𝜎2+2𝛼𝜀². (3.39) Let 𝐶₄=5𝜎/2+2/𝛼. By the Gronwall lemma, we have 𝑦(𝑡)≤𝑦𝑡₁−𝜎exp4𝑡−𝑡₁+4𝐶₄𝜀²𝜎. (3.40) Choosing 𝑡₂=𝑡₁+(4/𝜎)ln(𝜌²₁/𝜀²), it follows that 𝑦(𝑡)≤1+4𝐶₄𝜎𝜀²,𝑡≥𝑡₂. (3.41) That is, 𝐴𝑢₂−𝑓₂²+𝑣₂²≤𝐶𝜀², (3.42) for all 𝑡≥𝑡₂, where 𝐶=(1+4𝐶₄/𝜎).
Thus we complete the proof.

We are now in a position to state our main results as follows.

Theorem 3.6. Under the conditions (1.4), (1.5), (1.6), problem (1.1)–(1.3) has a global attractor 𝒜₁ in ℋ₁; it attracts all bounded subsets of ℋ₁ with respect to the norm of ℋ₁.

Proof. By Lemmas 3.1, 3.3, and 3.5, the conditions of Theorem 2.5 are satisfied. The proof is complete.

Corollary 3.7. The global attractor 𝒜₀ in ℋ₀ is coincides with 𝒜₁ in ℋ₁,thatis,𝒜₀=𝒜₁.

Proof. By Theorem 2.6, 𝒜₀ is a bounded set of ℋ₁, combining with Theorem 3.6, we can easily get 𝒜₀=𝒜₁.

Acknowledgment

The author of this paper would like to express her gratitude to Professor Chengkui Zhong for his guidance and encouragement, and to the reviewer, for valuable comments and suggestions. This work was supported in part by the Scientific Research Foundation of Graduate School of Nanjing University (2012CL21) and Taizhou Natural Science and Technology Development Project 2011.