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

Abstract

Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domain Ω. We prove the regularity and exponential growth of a pullback attractor in the space 𝑆²⁰(Ω)∩𝐿^2𝑝−2(Ω) 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 Ω be a bounded domain in ℝ^𝑁1×ℝ^𝑁2(𝑁₁,𝑁₂≥1), with smooth boundary 𝜕Ω. In this paper, we consider the following problem: 𝑢_𝑡−𝐺_𝑠𝑢+𝑢𝑓((𝑥𝑢𝑢,)𝑡()=𝑥=,𝑔𝜏0)(𝑡,=,𝑥𝑢𝑥)_𝜏,∈𝜕(Ω𝑡,,𝑥)𝑡∈∈𝑄(𝜏,_𝜏,𝑇𝑇=],(𝜏,𝑇]×Ω,(𝑥),𝑥∈Ω,(1.1) where 𝐺_𝑠𝑢=Δ_𝑥1𝑢+||𝑥₁||^2𝑠Δ_𝑥2𝑢,𝑥=𝑥₁,𝑥₂∈Ω⊂ℝ^𝑁1×ℝ^𝑁2,𝑠⩾0,(1.2) is the Grushin operator, 𝑢_𝜏∈𝐿²(Ω) is given, the nonlinearity 𝑓 and the external force 𝑔 satisfy the following conditions.(H1) The nonlinearity 𝑓∈𝐶¹(ℝ,ℝ) satisfies𝑓(𝑢||𝑓)′𝑢(𝑢≥)||𝐶≤₁|𝐶𝑢|^𝑝₃−𝐶₂,𝑝≥2,(1.3)𝑓′(𝑢)≥|𝑢|−^𝑝−2ℓ+,𝐶₄,((11..45)) where ℓ, 𝐶_𝑖, (𝑖=1,2,3,4) are positive constants. Relation (1.3) and (1.4) imply that𝛼₁||𝑢|^𝑝−𝛼₂≤𝐹(𝑢)≤𝛼₃||𝑢|^𝑝+𝛼₄,(1.6) where 𝐹(𝑠)=∫^𝑠0𝑓(𝜏)𝑑𝜏, and 𝛼_𝑖(𝑖=1,2,3,4) are positive constants.(H2)𝑔∈𝑊^1,2loc(ℝ;𝐿²(Ω)) satisfies^𝑡−∞𝑒^𝜆1𝑠‖𝑔(𝑠)‖^2𝐿2_(Ω)+‖𝑔′(𝑠)‖^2𝐿_2(Ω)𝑑𝑠<+∞,∀𝑡∈ℝ,(1.7) where 𝜆₁ is the first eigenvalue of the operator −𝐺_𝑠 in Ω with the homogeneous Dirichlet boundary condition.

The Grushin operator 𝐺_𝑠 was first introduced in [1]. Noting that if 𝑠>0, then 𝐺_𝑠 is not elliptic in domains of ℝ^𝑁1×ℝ^𝑁2 which intersect the hyperplane {𝑥₁=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(Ω)∩𝐿^𝑝(Ω) 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(Ω) (see Section 2 for its definition) and 𝐿^2𝑝−2(Ω). As we know, if the external force 𝑔 is only in 𝐿²(Ω), then solutions of problem (1.1) are at most in 𝐿^2𝑝−2(Ω)∩𝑆²⁰(Ω) 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(Ω)∩𝐿^𝑝(Ω), we only need assumption (H2) of the external force 𝑔; however, to prove the existence of pullback attractors in 𝑆²⁰(Ω) and 𝐿^2𝑝−2(Ω), 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(Ω)∩𝐿^2𝑝−2(Ω) 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(Ω) 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(Ω) and 𝐿^2𝑝−2(Ω) by using the asymptotic a priori estimate method. In Section 4, under additional assumptions of 𝑔, an exponential growth in 𝑆²⁰(Ω)∩𝐿^2𝑝−2(Ω) 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(Ω) defined as the completion of 𝐶^∞0(Ω) in the following norm: ‖𝑢‖_𝑆¹⁰_(Ω)=_Ω|||∇^𝑥1𝑢|||²+||𝑥₁||^2𝑠||∇_𝑥2𝑢||²𝑑𝑥^1/2(2.1)

and the scalar product ((𝑢,𝑣))∶=_Ω∇^𝑥₁𝑢∇_𝑥1𝑣+||𝑥₁||^2𝑠∇_𝑥2𝑢∇_𝑥2𝑣𝑑𝑥^1/2.(2.2)The following lemma comes from [16].

Lemma 2.1. Assume that Ω is a bounded domain in ℝ^𝑁1×ℝ^𝑁2(𝑁₁,𝑁₂≥0). Then the following embeddings hold:(i)𝑆¹⁰(Ω)↪𝐿^2∗𝛼(Ω) continuously;(ii)𝑆¹⁰(Ω)↪𝐿^𝑝(Ω) compactly if 𝑝∈[1,2^∗𝑠), where2^∗𝑠=2𝑁(𝑠)/(𝑁(𝑠)−2), 𝑁(𝑠)=𝑁₁+(𝑠+1)𝑁₂.

Now, we introduce the space 𝑆²⁰(Ω) defined as the closure of 𝐶^∞0(Ω) with the norm ‖𝑢‖_𝑆²⁰_(Ω)=_Ω||Δ^𝑥₁𝑢||²+||𝑥₁||^2s||Δ_𝑥2𝑢||²𝑑𝑥^1/2=_Ω||G_s𝑢||²𝑑𝑥^1/2.(2.3)

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

Lemma 2.2. Assume that Ω is a bounded domain in ℝ^𝑁1×ℝ^𝑁2(𝑁₁,𝑁₂≥0), with smooth boundary 𝜕Ω. Then 𝑆²⁰(Ω)⊂𝑆¹⁰(Ω) continuously.

It is known that (see, e.g., [3]) for the operator 𝐴=−𝐺_𝑠, there exist {𝑒_𝑗}_𝑗≥1

such that0<𝑒𝜆_𝑗1,𝑒_𝑘=𝛿_𝑗𝑘,𝐴𝑒_𝑗=𝜆_𝑗𝑒_𝑗,𝑗,𝑘=1,2,…,≤𝜆₂≤𝜆₃≤⋯,𝜆_𝑗⟶+∞as𝑗⟶∞,(2.4)

and {𝑒_𝑗}_𝑗≥1 is a complete orthonormal system in 𝐿²(Ω).

2.2. Pullback Attractors

Let 𝑋 be a Banach space with the norm ‖⋅‖. ℬ(𝑋) denotes all bounded sets of 𝑋. The Hausdorff semidistance between 𝐴 and 𝐵 is defined bydist(𝐴,𝐵)=sup_𝑥∈𝐴inf_𝑦∈𝐵‖𝑥−𝑦‖.(2.5)

Let {𝑈(𝑡,𝜏)∶𝑡≥𝜏,𝜏∈ℝ} be a process in 𝑋, that is, 𝑈(𝑡,𝜏)∶𝑋→𝑋 such that 𝑈(𝜏,𝜏)=𝐼𝑑 and 𝑈(𝑡,𝑠)𝑈(𝑠,𝜏)=𝑈(𝑡,𝜏) for all 𝑡≥𝑠≥𝜏, 𝜏∈ℝ. The process {𝑈(𝑡,𝜏)} is said to be norm-to-weak continuous if 𝑈(𝑡,𝜏)𝑥_𝑛⇀𝑈(𝑡,𝜏)𝑥, as 𝑥_𝑛→𝑥 in 𝑋, for all 𝑡≥𝜏, 𝜏∈ℝ. The following result is useful for proving the norm-to-weak continuity of a process.

Proposition 2.3 (see [9]). Let 𝑋,𝑌 be two Banach spaces, and let 𝑋^∗,𝑌^∗ be, respectively, their dual spaces. Suppose that 𝑋 is dense in 𝑌, the injection 𝑖∶𝑋→𝑌 is continuous, and its adjoint 𝑖^∗∶𝑌^∗→𝑋^∗ is dense, and {𝑈(𝑡,𝜏)} is a continuous or weak continuous process on 𝑌. Then {𝑈(𝑡,𝜏)} is norm-to-weak continuous on 𝑋 if and only if for 𝑡≥𝜏, 𝜏∈ℝ, 𝑈(𝑡,𝜏) maps compact sets of 𝑋 into bounded sets of 𝑋.

Definition 2.4. The process {𝑈(𝑡,𝜏)} is said to be pullback asymptotically compact if for any 𝑡∈ℝ, any 𝐷∈ℬ(𝑋), any sequence 𝜏_𝑛→−∞, and any sequence {𝑥_𝑛}⊂𝐷, the sequence {𝑈(𝑡,𝜏_𝑛)𝑥_𝑛} is relatively compact in 𝑋.

Definition 2.5. A family of bounded sets ℬ={𝐵(𝑡)∶𝑡∈ℝ}⊂𝑋 is called a pullback absorbing set for the process {𝑈(𝑡,𝜏)} if for any 𝑡∈ℝ and any 𝐷∈ℬ(𝑋), there exist 𝜏₀=𝜏₀(𝐷,𝑡)≤𝑡 and 𝐵(𝑡)∈ℬ such that _𝜏_≤𝜏0𝑈(𝑡,𝜏)𝐷⊂𝐵(𝑡).(2.6)

Definition 2.6. The family 𝒜={𝐴(𝑡)∶𝑡∈ℝ}⊂ℬ(𝑋) is said to be a pullback attractor for {𝑈(𝑡,𝜏)} if (1)𝐴(𝑡) is compact for all 𝑡∈ℝ, (2)𝒜 is invariant, that is, 𝑈(𝑡,𝜏)𝐴(𝜏)=𝐴(𝑡),∀𝑡≥𝜏,(2.7)(3)𝒜 is pullback attracting, that is, lim_{𝜏→−∞}dist(𝑈(𝑡,𝜏)𝐷,𝐴(𝑡))=0,∀𝐷∈ℬ(𝑋),andall𝑡∈ℝ,(2.8)(4)if {𝐶(𝑡)∶𝑡∈ℝ} is another family of closed pullback attracting sets, then 𝐴(𝑡)⊂𝐶(𝑡), for all 𝑡∈ℝ.

Theorem 2.7 (see [13]). Let {𝑈(𝑡,𝜏)} be a norm-to-weak continuous process which is pullback asymptotically compact. If there exists a pullback absorbing set ℬ={𝐵(𝑡)∶𝑡∈ℝ}, then {𝑈(𝑡,𝜏)} has a unique pullback attractor 𝒜={𝐴(𝑡)∶𝑡∈ℝ} and 𝐴(𝑡)=_𝑠≤𝑡_𝜏≤𝑠𝑈(𝑡,𝜏)𝐵(𝜏).(2.9)

In the rest of the paper, we denote by |⋅|₂, (⋅,⋅) the norm and inner product in 𝐿²(Ω), respectively, and by |⋅|_𝑝 the norm in 𝐿^𝑝(Ω). By ‖⋅‖ we denote the norm in 𝑆¹⁰(Ω). For a Banach space 𝐸, ‖⋅‖_𝐸 will be the norm. We also denote by 𝐶 an arbitrary constant, which is different from line to line, and even in the same line.

3. Existence of Pullback Attractors in 𝑆²⁰(Ω)∩𝐿^2𝑝−2(Ω)

It is well known (see, e.g., [2] or [14]) that under conditions (𝐻1)−(𝐻2), problem (1.1) defines a process𝑈(𝑡,𝜏)∶𝐿²(Ω)⟶𝑆¹⁰(Ω)∩𝐿^𝑝(Ω),∀𝑡≥𝜏,(3.1)where 𝑈(𝑡,𝜏)𝑢_𝜏 is the unique weak solution of (1.1) with initial datum 𝑢_𝜏 at time 𝜏. The process {𝑈(𝑡,𝜏)} has a pullback attractor in 𝑆¹⁰(Ω)∩𝐿^𝑝(Ω).

In this section, we will prove that the pullback attractor is in fact in 𝑆²⁰(Ω)∩𝐿^2𝑝−2(Ω).

Lemma 3.1. Assuming that 𝑓 and 𝑔 satisfy (H 1)-(H 2), 𝑢(𝑡) is a weak solution of (1.1). Then the following inequality holds for 𝑡>𝜏: ‖𝑢‖²+|𝑢|^𝑝𝑝≤𝐶𝑒^{−𝜆1(𝑡−𝜏)}||𝑢_𝜏||²²+1+𝑒^−𝜆1𝑡^𝑡−∞𝑒^𝜆1𝑠||𝑔(𝑠)||²²𝑑𝑠,(3.2) where 𝐶 is a positive constant.

Proof. Multiplying (1.1) by 𝑢 and then integrating over Ω, we get 12𝑑𝑑𝑡|𝑢|²²+‖𝑢‖²+_Ω𝑓(𝑢)𝑢𝑑𝑥=_Ω𝑔(𝑡)𝑢𝑑𝑥≤1𝜆₁||𝑔(𝑡)||²²+𝜆¹4|𝑢|²².(3.3) Using hypothesis (H1) and the inequality ‖𝑢‖²≥𝜆₁|𝑢|²², we have 𝑑𝑑𝑡|𝑢|²²+𝜆₁|𝑢|²²+𝐶‖𝑢‖²+|𝑢|^𝑝𝑝≤𝐶1+||𝑔(𝑡)||²².(3.4) Letting 𝐹(𝑠)=∫^𝑠0𝑓(𝜏)𝑑𝜏, by (H1), we have 𝛼₁|𝑢|^𝑝−𝛼₂≤𝐹(𝑢)≤𝛼₃|𝑢|^𝑝+𝛼₄.(3.5) Now multiplying (3.4) by 𝑒^𝜆1𝑡 and using (3.5), we get 𝑑𝑑𝑡𝑒^𝜆1𝑡||𝑢(𝑡)||²²+𝐶𝑒^𝜆1𝑡‖𝑢‖²+2_Ω𝐹(𝑢(𝑡))𝑑𝑥≤𝐶𝑒^𝜆1𝑡1+||𝑔(𝑡)||²².(3.6) Integrating (3.6) from 𝜏 to 𝑠∈[𝜏,𝑡−1] and 𝑠 to 𝑠+1, respectively, we obtain 𝑒^𝜆1𝑠||𝑢(𝑠)||²²≤𝐶𝑒^𝜆1𝜏||𝑢_𝜏||²²+𝐶𝑒^𝜆1𝑠+𝐶^𝑠𝜏𝑒^𝜆1𝑟||𝑔(𝑟)||²²𝑑𝑟,∀𝑠∈[𝜏,𝑡−1],(3.7)_𝑠𝑠+1𝑒^𝜆1𝑟‖𝑢(𝑟)‖²≤𝑒_𝜆1𝑠+2_Ω𝐹(𝑢(𝑥,𝑟))𝑑𝑥𝑑𝑟||𝑢(𝑠)||²²+𝐶^𝑠𝑠+1𝑒^𝜆1𝑟≤𝑒_𝜆1𝜏1+||𝑔(𝑟)||²²𝑑𝑟+||𝐶𝑢_𝜏𝑒||²²+𝐶_{𝜆1(𝑠+1)}𝑒^𝜆1𝑠+𝐶^𝑠𝜏𝑒^𝜆1𝑟||𝑔(𝑟)||²²𝑑𝑟≤𝐶𝑒_𝜆1𝜏+C^𝑠𝑠+1𝑒^𝜆1𝑟||𝑔(𝑟)||²²𝑑𝑟||𝑢_𝜏||²²+𝑒^𝜆1𝑡+^𝑡𝜏𝑒^𝜆1𝑟||𝑔(𝑟)||²²𝑑𝑟.(3.8) Multiplying (1.1) by 𝑢_𝑡 and integrating over Ω, we have ||𝑢_𝑡(𝑠)||²²+12𝑑𝑑𝑠‖𝑢(𝑠)‖²+2_Ω𝐹(𝑢(𝑥,𝑠))𝑑𝑥=_Ω𝑔(𝑠)𝑢_𝑡(𝑠)≤12||𝑔(𝑠)||²²+12||𝑢^𝑡(𝑠)||²².(3.9) Thus 𝑒^𝜆1𝑠||𝑢_𝑡(𝑠)||²²+𝑑𝑑𝑠𝑒^𝜆1𝑠‖𝑢(𝑠)‖²≤𝜆₁+2∫_Ω𝐹(𝑢(𝑥,𝑠))𝑑𝑥𝑒^𝜆1𝑠‖𝑢(𝑠)‖²+2_Ω𝐹(𝑢(𝑥,𝑠))𝑑𝑥+𝑒^𝜆1𝑠||𝑔(𝑠)||²².(3.10) Combining (3.8) and (3.10), and using the uniform Gronwall inequality, we have 𝑒^𝜆1𝑡‖𝑢(𝑡)‖²+2_Ω𝐹(𝑢(𝑥,𝑡))𝑑𝑥≤𝐶𝑒^𝜆1𝜏||𝑢_𝜏||²+𝑒^𝜆1𝑡+^𝑡−∞𝑒^𝜆1𝑠||𝑔(𝑠)||²²𝑑𝑠.(3.11) Using (H1) once again and thanks to 𝑒^𝜆1𝜏|𝑢_𝜏|²²→0 as 𝜏→−∞, we get the desired result from (3.11).

Lemma 3.2. Assume that (H 1), (H 2) hold. Then for any 𝑡∈ℝ and any 𝐷⊂𝐿²(Ω) that is bounded, there exists 𝜏₀≤𝑡−1 such that ||𝑢_𝑡(𝑡)||²²≤𝐶1+𝑒^−𝜆1𝑡^𝑡−∞𝑒^𝜆1𝑠||𝑔(𝑠)||²²+||𝑔′(𝑠)||²²𝑑𝑠,(3.12) for any 𝜏≤𝜏₀ and any 𝑢_𝜏∈𝐷, where 𝑢_𝑡(𝑠)=(𝑑/𝑑𝑡)(𝑈(𝑡,𝜏)𝑢_𝜏)|_𝑡=𝑠.

Proof. Integrating (3.10) from 𝑟 to 𝑟+1,𝑟∈[𝜏,𝑡−1] and using (3.8) and (3.11), in particular we find ^𝑟𝑟+1𝑒^𝜆1s||𝑢_𝑡||²²𝑑𝑠≤𝑒^𝜆1𝑟‖𝑢(𝑟)‖²+𝜆₁+2_Ω𝐹(𝑢(𝑥,𝑟))𝑑𝑥^𝑟𝑟+1𝑒^𝜆1𝑠‖𝑢(𝑠)‖²++2_Ω𝐹(𝑢(𝑥,𝑠))𝑑𝑥𝑑𝑠_𝑟𝑟+1𝑒^𝜆1𝑠||𝑔(𝑠)||²²𝑑𝑠≤𝐶𝑒^𝜆1𝜏||𝑢_𝜏||²²+𝑒^𝜆1𝑡+^𝑡−∞𝑒^𝜆1𝑠||𝑔(𝑠)||²²𝑑𝑠.(3.13) On the other hand, differentiating (1.1) and denoting  𝑣=𝑢_𝑡, we have 𝑣_𝑡−𝐺_𝑠𝑣+𝑓^(𝑢)𝑣=𝑔^(𝑡).(3.14) Taking the inner product of (3.14) with 𝑣 in 𝐿²(Ω), we get 12𝑑𝑑𝑡|𝑣|²²+‖𝑣‖²+(𝑓′(𝑢)𝑣,𝑣)=𝑔^(𝑡),𝑣.(3.15) Using (1.5) and Young’s inequality, after a few computations, we see that 𝑑𝑑𝑡𝑒^𝜆1𝑡|𝑣|²²+2𝑒^𝜆1𝑡‖𝑣‖²≤𝐶𝑒^𝜆1𝑡|𝑣|²²+𝐶𝑒^𝜆1𝑡||𝑔^(𝑡)||²².(3.16) Combining (3.16) and (3.13) and using the uniform Gronwall inequality, we obtain 𝑒^𝜆1𝑡||𝑣(𝑡)||²²≤𝐶𝑒^𝜆1𝜏||𝑢_𝜏||²²+𝑒^𝜆1𝑡+^𝑡−∞𝑒^𝜆1𝑠||𝑔(𝑠)||²²+||𝑔^(𝑠)||²²𝑑𝑠.(3.17) The proof is now complete because 𝑒^𝜆1𝜏|𝑢_𝜏|²²→0 as 𝜏→−∞.

3.1. Existence of a Pullback Attractor in𝐿^2𝑝−2(Ω)

In this section, following the general lines of the method introduced in [9], we prove the existence of a pullback attractor in 𝐿^2𝑝−2(Ω). In order to do this, we need an additional condition of 𝑔^𝑡−∞𝑒^𝜆1𝑡‖‖𝑔^(𝑡)‖‖^𝑚′_𝐿𝑚(Ω)𝑑𝑡<+∞,∀t∈ℝ,(3.18) where 𝑚, 𝑚^ are defined as in (3.30).

Lemma 3.3. The process {𝑈(𝑡,𝜏)} associated to problem (1.1) has a pullback absorbing set in 𝐿^2𝑝−2(Ω).

Proof. Multiplying (1.1) by |𝑢|^p−2𝑢 and integrating over Ω, we get _Ω𝑢_𝑡=|𝑢|^𝑝−2𝑢𝑑𝑥+(𝑝−1)_Ω||∇^𝑥₁𝑢||²+||𝑥₁||^2s||∇_𝑥2𝑢||²|𝑢|^𝑝−2𝑑𝑥+_Ω𝑓(𝑢)|||𝑢|^𝑝−2𝑢𝑑𝑥_Ω||||𝑔(𝑡)𝑢||^𝑝−2𝑢𝑑𝑥.(3.19) From (1.3) and the fact that 𝐿^𝑝(Ω)⊂𝐿^𝑝−2(Ω) continuously, we have _Ω𝑓(𝑢)|𝑢|^𝑝−2𝑢𝑑𝑥≥_Ω𝐶₁|𝑢|^𝑝−𝐶₂|𝑢|^𝑝−2𝑑𝑥≥𝐶₁‖𝑢‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}−𝐶₂|𝑢|^𝑝𝑝.(3.20) On the other hand, by Cauchy’s inequality, we see that ||||_Ω𝑢_𝑡|𝑢|^𝑝−2𝑢𝑑𝑥||||≤𝐶¹4‖𝑢‖^{𝐿2𝑝−22𝑝−2(Ω)}+1𝐶₁||𝑢_𝑡||²²,(3.21)||||_Ω𝑔(𝑡)|𝑢|^𝑝−2𝑢𝑑𝑥||||≤𝐶¹4‖𝑢‖^{𝐿2𝑝−22𝑝−2(Ω)}+14||𝑔(𝑡)||²².(3.22) Combining (3.19)–(3.22) imply that ‖𝑢‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}≤𝐶||𝑢_𝑡||²²+|𝑢|^𝑝𝑝+||𝑔(𝑡)||²².(3.23) Applying (3.2) and Lemma 3.2, we conclude the existence of a pullback absorbing set in 𝐿^2𝑝−2(Ω) for the process 𝑈(𝑡,𝜏).

Lemma 3.4. For any 𝑠∈ℝ, any 2≤𝑝<∞, and any bounded set 𝐵⊂𝐿²(Ω), there exists 𝜏₀ such that _Ω||𝑢_𝑡(𝑠)||^𝑝𝑑𝑥≤𝑀,∀𝜏≤𝜏₀,𝑢_𝜏∈𝐵,(3.24) where 𝑀 depends on 𝑠, 𝑝 but not on 𝐵, and 𝑢_𝑡(𝑠)=(𝑑/𝑑𝑡)(𝑈(𝑡,𝜏)𝑢_𝜏)|_𝑡=𝑠.

Proof. We will prove the lemma by induction argument. Letting  𝛽=𝑁(𝑠)/(𝑁(𝑠)−2)>1 and denoting 𝑣=𝑢_𝑡 we prove that for 𝑘=0,1,2,…, there exist 𝜏_𝑘 and 𝑀_𝑘(𝑠) such that 𝑒^𝜆1𝑠_Ω_𝑠𝑠+1||𝑣(𝑠)||^2𝛽𝑘𝑑𝑥≤𝑀_𝑘(s)forany𝑢_𝜏∈𝐵,𝜏≤𝜏_𝑘,(𝑃_𝑘)𝑒^𝜆1𝑟_Ω||𝑣(𝑟)||^2𝛽𝑘+1𝑑𝑥^1/𝛽𝑑𝑟≤𝑀_𝑘(𝑠)forany𝑢_𝜏∈𝐵,𝜏≤𝜏_𝑘,(𝑄_𝑘) where 𝜏_𝑘 depends on 𝑘 and 𝐵 and 𝑀_𝑘 depends only on 𝑘.
For 𝑘=0, we have (𝑃₀) from (3.17). Integrating (3.16) and using 𝑆¹⁰(Ω)↪𝐿^2𝛽(Ω) continuously, we get (𝑄₀).
Assuming that (𝑃_𝑘), (𝑄_𝑘) hold, we prove so are (𝑃_𝑘+1) and (𝑄_𝑘+1). Multiplying (3.14) by |𝑣|^{2𝛽𝑘+1−2}𝑣 and integrating over Ω, we obtain𝐶𝑑𝑑𝑡_Ω≤ℓ|𝑣|^2𝛽𝑘+1_Ω𝑑𝑥+𝐶_Ω||∇^𝑥₁𝑣||²+||𝑥₁||^2𝑠||∇_𝑥2𝑣||²|𝑣|^{𝛽2𝑘+1−2}𝑑𝑥|𝑣|^2𝛽𝑘+1𝑑𝑥+𝑔′(𝑡),|𝑣|^{2𝛽𝑘+1−2}𝑣.(3.25) Using the imbedding 𝑆¹⁰(Ω)↪𝐿^2𝛽(Ω) once again, we get _Ω||∇^𝑥₁=‖‖𝑣𝑣||²+||𝑥₁||^2𝑠||∇_𝑥2𝑣||²|𝑣|^𝛽2𝑘+1𝑑𝑥≥_Ω|𝑣|²|𝑣|^{2𝛽2𝑘+1−2}𝑑𝑥_𝛽𝑘+1‖‖^2𝐿_2𝛽(Ω)=_Ω|𝑣|^2𝛽𝑘+2𝑑𝑥^1/𝛽.(3.26) Combining Holder’s and Young’s inequalities, we see that _Ω𝑔^(𝑡)|𝑣|^{2𝛽𝑘+1−2}𝑣𝑑𝑥≤_Ω||𝑔^(𝑡)||^𝑚𝑑𝑥^1/𝑚_Ω|𝑣|^{(2𝛽𝑘+1−1)𝑛}𝑑𝑥^1/𝑛≤∫^Ω||𝑔^(𝑡)||^𝑚𝑑𝑥^𝑚′/𝑚𝑚^+∫^Ω|𝑣|^{(2𝛽𝑘+1−1)𝑛}𝑑𝑥^𝑛′/𝑛𝑛^,(3.27) where 1/𝑚+1/𝑛=1/𝑚^+1/𝑛^=1. Choose 𝑛, 𝑛^ such that 2𝛽^𝑘+1−1𝑛=2𝛽^𝑘+2,𝑛𝑛^=𝛽1,(3.28) thus 𝑛=2𝛽^𝑘+22𝛽^𝑘+1−1,𝑛^=2𝛽^𝑘+12𝛽^𝑘+1−1.(3.29) Hence 𝑚=𝑛𝑛−1=2𝛽^𝑘+1−12𝛽^𝑘+2−2𝛽^𝑘+1+1,𝑚^=2𝛽^𝑘+1.(3.30) Then from (3.27), we infer that _Ω𝑔^(𝑡)|𝑣|^{2𝛽𝑘+1−2}𝑣𝑑𝑥≤1𝑚^‖‖𝑔^(𝑡)‖‖^𝑚′_𝐿𝑚(Ω)+1𝑛^_Ω|𝑣|^2𝛽𝑘+2𝑑𝑥^1/𝛽.(3.31) Applying (3.26) and (3.31) in (3.25), we find that 𝑑𝑑𝑡𝑒^𝜆1𝑡_Ω|𝑣|^2𝛽𝑘+1𝑑𝑥+𝐶𝑒^𝜆1𝑡_Ω|𝑣|^2𝛽𝑘+2𝑑𝑥^1/𝛽≤𝐶𝑒^𝜆1𝑡_Ω|𝑣|^2𝛽𝑘+1𝑑𝑥+𝐶𝑒^𝜆1𝑡‖‖𝑔^(𝑡)‖‖^𝑚′_𝐿𝑚(Ω).(3.32) Combining (𝑄_𝑘) and (3.32), using the uniform Gronwall inequality and taking into account assumption (3.18), we get (𝑃_𝑘+1). On the other hand, integrating (3.32) from 𝑡 to 𝑡+1, we find (𝑄_𝑘+1). Now since 𝛽>1, and taking 𝑘≥log_𝛽𝑝/2, we get the desired estimate.

We will use the following lemma.

Lemma 3.5 (see [15]). If there exists 𝜎>0 such that ∫^𝑡−∞𝑒^𝜎𝑠|𝜑(𝑠)|²𝑑𝑠<∞, for all 𝑡∈ℝ, then lim_𝛾→+∞^𝑡−∞𝑒^{−𝛾(𝑡−𝑠)}||𝜑(𝑠)||²𝑑𝑠=0,𝑡∈ℝ.(3.33)

Let 𝐻_𝑚=span{𝑒₁,𝑒₂,…,𝑒_𝑚} in 𝐿²(Ω), and let 𝑃_𝑚∶𝐿²(Ω)→𝐻_𝑚 be the orthogonal projection, where {𝑒_𝑖}^∞𝑖=1 are the eigenvectors of operator 𝐴=−𝐺_𝑠. For any 𝑢∈𝐿²(Ω), we write𝑢=𝑃_𝑚𝑢+𝐼−𝑃_𝑚𝑢=𝑢₁+𝑢₂.(3.34)

Lemma 3.6. For any 𝑡∈ℝ, any 𝐵⊂𝐿²(Ω) and any 𝜀, there exist 𝜏₀(𝑡,𝐵,𝜀) and 𝑚₀∈ℕ such that ||𝐼−𝑃_𝑚𝑣||²²<𝜀,∀𝜏≤𝜏₀,∀𝑢_𝜏∈𝐵,𝑚≥𝑚₀.(3.35)

Proof. Multiplying (3.14) by 𝑣₂=(𝐼−𝑃_𝑚)𝑣 and then integrating over Ω, using |∇𝑣₂|²²≥𝜆_𝑚|𝑣₂|²² and Cauchy’s inequality we get 𝑑𝑑𝑡||𝑣²||²²+𝜆_𝑚||𝑣₂||²²≤𝐶_Ω||𝑓^(𝑢)𝑣||²𝑑𝑥+𝐶||𝑔^(𝑡)||²².(3.36) We multiply (3.36) by 𝑒^𝜆𝑚𝑡 and use assumption (1.4). We get 𝑑𝑑𝑡𝑒^𝜆𝑚𝑡||𝑣₂||²²≤𝐶𝑒^𝜆𝑚𝑡_Ω|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥+𝐶𝑒^𝜆𝑚𝑡||𝑔^(𝑡)||²².(3.37) Integrating (3.37) from 𝑠 to 𝑡, 𝑒^𝜆𝑚𝑡||𝑣₂(𝑡)||²²≤𝑒^𝜆𝑚𝑠||𝑣₂(𝑠)||²²+𝐶^𝑡𝑠𝑒^𝜆𝑚𝑟_Ω≤𝑒_𝜆𝑚𝑠|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥𝑑𝑟+𝐶^𝑡𝑠𝑒^𝜆𝑚𝑟||𝑔′(𝑟)||²²𝑑𝑟||𝑣(𝑠)||²²+𝐶^𝑡−∞𝑒^𝜆𝑚𝑟_Ω|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥𝑑𝑟+𝐶^𝑡−∞𝑒^𝜆𝑚𝑟||𝑔^(𝑟)||²²𝑑𝑟.(3.38) Now integrating (3.38) with respect to 𝑠 from 𝜏 to 𝑡, we infer that(𝑡−𝜏)𝑒^𝜆𝑚𝑡||𝑣₂(𝑡)||²²≤^𝑡𝜏𝑒^𝜆𝑚𝑟||𝑣(𝑟)||²²𝑑𝑟+𝐶(𝑡−𝜏)^𝑡−∞𝑒^𝜆𝑚𝑟_Ω+𝐶(𝑡−𝜏)_𝑡−∞|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥𝑑𝑟𝑒^𝜆𝑚𝑟||𝑔^(𝑟)||²²𝑑𝑟.(3.39) Thus ||𝑣₂(𝑡)||²²≤1𝑡−𝜏^𝑡−∞𝑒^{−𝜆𝑚(𝑡−𝑟)}||𝑣(𝑟)||²²𝑑𝑟+𝐶^𝑡−∞𝑒^{−𝜆𝑚(𝑡−𝑟)}_Ω+𝐶_𝑡−∞|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥𝑑𝑟𝑒^{−𝜆𝑚(𝑡−𝑟)}||𝑔^(𝑟)||²²𝑑𝑟.(3.40) By Lemma 3.5 and since 𝜆_𝑚→+∞ as 𝑚→+∞, there exist 𝜏₁ and 𝑚₁ such that 1𝑡−𝜏^𝑡−∞𝑒^{−𝜆𝑚(𝑡−𝑟)}𝐶^𝑡−∞𝑒^{−𝜆𝑚(𝑡−𝑟)}||𝑔^(𝑟)||²²𝑑𝑟<𝜀||𝑣(𝑟)||₂₂3,𝑑𝑟<𝜀3,(3.41) for all 𝜏≤𝜏₁ and 𝑚≥𝑚₁. For the second term of the right-hand side of (3.40), using Holder’s inequality we have ^𝑡−∞𝑒^{−𝜆𝑚(𝑡−𝑟)}_Ω≤|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥𝑑𝑟_𝑡−∞_Ω𝑒^{((−𝑝−1)/(𝑝−2)𝜆𝑚)(𝑡−𝑟)}|𝑢|^2𝑝−2𝑑𝑥^{(𝑝−2)/(𝑝−1)}_Ω≤𝑒^{−(𝑝−1)𝜆𝑚(𝑡−𝑟)}|𝑣|^2𝑝−2𝑑𝑥^1/(𝑝−1)𝑑𝑟_𝑡−∞𝑒^{((−𝑝−1)/(𝑝−2)𝜆𝑚)(𝑡−𝑟)}‖𝑢‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}𝑑𝑟^{(𝑝−2)/(𝑝−1)}^𝑡−∞𝑒^{−(𝑝−1)𝜆𝑚(𝑡−𝑟)}𝑑𝑟_Ω|𝑣|^2𝑝−2𝑑𝑥^1/(𝑝−1).(3.42) From Lemmas (3.5)–(3.7), we see that there exist 𝜏₂ and 𝑚₂∈ℕ such that 𝐶^𝑡−∞𝑒^{−𝜆𝑚(𝑡−𝑟)}_Ω|𝑢|^2(𝑝−2)|𝑣|²𝑑𝑥𝑑𝑟<𝜀3,∀𝜏≤𝜏⁰,𝑚≥𝑚₂.(3.43) Let 𝜏₀=min{𝜏₁,𝜏₂} and 𝑚₀=max{𝑚₁,𝑚₂}, from (3.40), taking into account (3.41) and (3.43), we obtain (3.35).

Lemma 3.7 (see [9]). Let 𝐵 be a bounded subset in 𝐿^𝑞(Ω)(𝑞≥1). If 𝐵 has a finite 𝜀-net in 𝐿^𝑞(Ω), then there exists an 𝑀=𝑀(𝐵,𝜀), such that for any 𝑢∈𝐵, the following estimate is valid: _{Ω(|𝑢|≥𝑀)}|𝑢|^𝑞𝑑𝑥<𝜀.(3.44)

Using Lemma 3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set {𝑢_𝑡(𝑠)∶𝑠≤𝑡,𝑢_𝜏∈𝐵} has a finite 𝜖-net in 𝐿²(Ω). Therefore, we get the following result.

Lemma 3.8. For any 𝑡∈ℝ, any 𝐵⊂𝐿²(Ω) that is bounded, and any 𝜀>0, there exists 𝜏₀≤𝑡 and 𝑀₀>0 such that_{Ω(|𝑢|≥𝑀)}||𝑢_𝑡(𝑡)||²𝑑𝑥<𝜀,∀𝜏<𝜏₀,𝑀>𝑀₀,𝑢_𝜏∈𝐵.(3.45)

Lemma 3.9 (see [9]). For any 𝑡∈ℝ, any bounded set 𝐵⊂𝐿²(Ω), and any 𝜀>0, there exist 𝜏₀ and 𝑀₀>0 such that mes(Ω(𝑢(𝑡)≥𝑀))<𝜀∀𝜏≤𝜏₀,𝑀≥𝑀₀,𝑢_𝜏∈𝐵,(3.46) where mes is the Lebesgue measure in ℝ^𝑁 and Ω(𝑢(𝑡)≥𝑀)={𝑥∈Ω∶𝑢(𝑡,𝑥)≥𝑀}.

Lemma 3.10 (see [2]). Let {𝑈(𝑡,𝜏)} be a norm-to-weak continuous process in 𝐿²(Ω) and 𝐿^𝑞(Ω),𝑞≥2. Then {𝑈(𝑡,𝜏)} is pullback asymptotically compact in 𝐿^𝑞(Ω) if
(i){𝑈(𝑡,𝜏)} is pullback asymptotically compact in 𝐿²(Ω); (ii)for any 𝑡∈ℝ, any bounded set 𝐷⊂𝐿²(Ω), and any 𝜖>0, there exist 𝑀>0 and𝜏₀≤𝑡sup_𝜏≤𝜏0sup_{𝑢𝜏∈𝐷}_{Ω(|𝑈(𝑡,𝜏)𝑢𝜏|≥𝑀)}||𝑈(𝑡,𝜏)𝑢_𝜏||^𝑞𝑑𝑥≤𝐶𝜖,(3.47) where 𝐶 is independent of 𝑀, 𝜏, 𝑢_𝜏, and 𝜖.

We are now ready to prove the existence of a pullback attractor in 𝐿^2𝑝−2(Ω).

Theorem 3.11. Assume that assumptions (1.3)–(1.7) and (3.18) hold. Then the process {𝑈(𝑡,𝜏)} associated to problem (1.1) possesses a pullback attractor 𝒜_2𝑝−2={𝐴_2𝑝−2(𝑡)}_𝑡∈ℝ in 𝐿^2𝑝−2(Ω).

Proof. Because of Lemma 3.10, since {𝑈(𝑡,𝜏)} has a pullback absorbing set in 𝐿^2𝑝−2(Ω), we only have to prove that for any 𝑡∈ℝ, any 𝐵⊂𝐿²(Ω), and any 𝜀>0, there exist 𝜏₂≤𝑡 and 𝑀₂>0 such that _{Ω(|𝑢|≥𝑀)}|𝑢|^2𝑝−2𝑑𝑥≤𝐶𝜀,∀𝜏≤𝜏₂,𝑀≥𝑀₂,𝑢_𝜏∈𝐵.(3.48) Taking the inner product of (1.1) with (𝑢−𝑀)^+𝑝−1 in 𝐿²(Ω), where (𝑢−𝑀)₊=𝑢0−𝑀iiff𝑢𝑢≥<𝑀𝑀,,(3.49) we have _Ω𝑢+_𝑡(𝑢−𝑀)^+𝑝−1𝑑𝑥+(𝑝−1)_Ω||∇^𝑥₁𝑢||²𝑑𝑥+||𝑥₁||^2𝑠||∇_𝑥2𝑢||²(𝑢−𝑀)^+𝑝−2𝑑𝑥_Ω≤𝑓(𝑢)(𝑢−𝑀)^+𝑝−1𝑑𝑥_Ω𝑔(𝑡)(𝑢−𝑀)^+𝑝−1𝑑𝑥.(3.50) Some standard computations give us _Ω𝑓(𝑢)(𝑢−𝑀)^+𝑝−1𝑑𝑥≥𝐶_{Ω(𝑢≥𝑀)}|𝑢|^2𝑝−2𝑑𝑥+𝐶_{Ω(𝑢≥𝑀)}|𝑢|^𝑝𝑑𝑥,(3.51)−_Ω𝑢_𝑡(𝑢−𝑀)^+𝑝−1𝑑𝑥≤𝐶4_{Ω(𝑢≥𝑀)}|𝑢|^2𝑝−2𝑑𝑥+1𝐶_{Ω(𝑢≥𝑀)}||𝑢_𝑡||²𝑑𝑥,(3.52)_Ω𝑔(𝑡)(𝑢−𝑀)^+𝑝−1𝑑𝑥≤𝐶4_{Ω(𝑢≥𝑀)}|𝑢|^2𝑝−2d𝑥+1𝐶_{Ω(𝑢≥𝑀)}||𝑔(𝑡)||²𝑑𝑥.(3.53) Combining (3.50)–(3.53), we find _{Ω(𝑢≥𝑀)}|𝑢|^2𝑝−2𝑑𝑥≤𝐶_{Ω(𝑢≥𝑀)}||𝑢_𝑡||²𝑑𝑥+_{Ω(𝑢≥𝑀)}||𝑔(𝑡)||²𝑑𝑥+𝐶_{Ω(𝑢≥𝑀)}|𝑢|^𝑝𝑑𝑥.(3.54) Applying Lemmas 3.7 and 3.8 to (3.54) we find there exist 𝜏₀ and 𝑀₀ such that _{Ω(𝑢≥𝑀)}|𝑢|^2𝑝−2𝑑𝑥<𝜀∀𝜏≤𝜏₀,𝑀≥𝑀₀.(3.55) Repeating the above arguments with |(𝑢+𝑀)₋|^𝑝−2(𝑢+𝑀)₋ in place of (𝑢−𝑀)^+𝑝−1, we have _{Ω(𝑢≤−𝑀)}|𝑢|^2𝑝−2𝑑𝑥<𝜀∀𝜏≤𝜏₁,𝑀≥𝑀₁,(3.56) for some 𝜏₁≤𝑡 and 𝑀₁>0, where (𝑢+𝑀)₋=𝑢0+𝑀iiff𝑢𝑢>≤𝑀−.𝑀,(3.57) Letting 𝜏₂=min{𝜏₀,𝜏₁} and 𝑀₂=max{𝑀₀,𝑀₁} we have _{Ω(|𝑢|≥𝑀2)}|𝑢|^2𝑝−2𝑑𝑥<𝐶𝜀,∀𝜏≤𝜏₂,𝑀≥𝑀₂.(3.58) This completes the proof.

3.2. Existence of a Pullback Attractor in 𝑆²⁰(Ω)

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

Lemma 3.12. The process {𝑈(𝑡,𝜏)} associated to (1.1) has a pullback absorbing set in 𝑆²⁰(Ω).

Proof. We multiply (1.1) by −𝐺_𝑠𝑢; then, using 𝑓(0)=0, we have ‖𝑢‖^2𝑆_20(Ω)=_Ω𝑢_𝑡𝐺_𝑠𝑢𝑑𝑥−_Ω𝑓′(𝑢)||∇_𝑥1𝑢||²+||𝑥₁||^2𝑠||∇_𝑥2𝑢||²𝑑𝑥−_Ω𝑔(𝑡)𝐺_𝑠𝑢𝑑𝑥.(3.59) Using 𝑓′(𝑢)≥−ℓ, Cauchy’s inequality, and argument as in Lemma 3.3, from (3.59) we have ‖𝑢‖^2𝑆_20(Ω)≤2||𝑢_𝑡||²²+ℓ‖𝑢‖²+||𝑔(𝑡)||²².(3.60) Taking into account (3.11), the proof is complete.

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

Definition 3.13. A process {𝑈(𝑡,𝜏)} is said to satisfy (PDC) condition in 𝑋 if for any 𝑡∈ℝ, any bounded set 𝐵⊂𝐿²(Ω) and any 𝜀>0, there exists 𝜏₀≤𝑡 and a finite dimensional subspace 𝑋₁ of 𝑋 such that (i)𝑃(⋃_𝜏≤𝜏0𝑈(𝑡,𝜏)𝐵) is bounded in 𝑋; and(ii)‖(𝐼_𝑋−𝑃)𝑈(𝑡,𝜏)𝑢_𝜏‖_𝑋<𝜀, for all 𝜏≤𝜏₀ and 𝑢_𝜏∈𝐵, where 𝑃∶𝑋→𝑋₁ is a canonical projection and 𝐼_𝑋 is the identity.

Lemma 3.14 (see [13]). If a process {𝑈(𝑡,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 𝐴⊂𝐿^2𝑝−2(Ω), if 𝜅(𝐴)<𝜀 in 𝐿^2𝑝−2(Ω), then we have 𝜅(𝑓(𝐴))<𝐶𝜀in𝐿²(Ω),(3.61) where the Kuratowski noncompactness measure 𝜅(𝐵) in a Banach space 𝑋 defined as 𝜅(𝐵)=inf{𝛿∶𝐵hasafinitecoveringbyballsin𝑋withradii𝛿}.(3.62)

Theorem 3.16. Assume that 𝑓 satisfies (1.3)–(1.5), 𝑔 satisfies (1.7) and (3.18). Then the process {𝑈(𝑡,𝜏)} generated by (1.1) has a pullback attractor 𝒜={𝐴(𝑡)∶𝑡∈ℝ} in 𝑆²⁰(Ω).

Proof. We consider a complete trajectory 𝑢(𝑡) lies on pullback attractor 𝒜_2𝑝−2 in 𝐿^2𝑝−2(Ω) for 𝑈(𝑡,𝜏), that is, 𝑢(𝑡)∈𝒜_2𝑝−2(𝑡) and 𝑈(𝑡,𝜏)𝑢_𝜏=𝑢(𝑡), for all 𝑡≥𝜏. Denoting 𝐴=−𝐺_𝑠 and multiplying (1.1) by 𝐴𝑢₂=𝐴(𝐼−𝑃_𝑚)𝑢=(𝐼−𝑃_𝑚)𝐴𝑢 we have 𝑢_𝑡,𝐴𝑢₂+‖‖𝑢₂‖‖^2𝑆_20(Ω)+_Ω𝑓(𝑢)𝐴𝑢₂𝑑𝑥=𝑔(𝑡),𝐴𝑢₂.(3.63) Using Holder’s inequality we get ‖‖𝑢₂‖‖^2𝑆_20(Ω)≤𝐶||(𝐼−𝑃_𝑚)𝑢_𝑡||²²+||𝐼−𝑃_𝑚𝑔(𝑡)||²²+_Ω(𝑓(𝑢))²𝑑𝑥.(3.64) Thanks to Lemmas 3.6 and 3.15 and the fact that 𝑔∈𝐶_loc(ℝ;𝐿²(Ω)), we see that {𝑈(𝑡,𝜏)} satisfies condition (PDC) in 𝑆²⁰(Ω). Now from Lemmas 3.3 and 3.14 we get the desired result.

4. Exponential Growth in 𝑆²⁰(Ω)∩𝐿^2𝑝−2(Ω) of Pullback Attractors

In this section, we will give an exponential growth condition in 𝑆²⁰(Ω)∩𝐿^2𝑝−2(Ω) for the pullback attractor 𝒜(𝜏).

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

Lemma 4.1. Let 𝑋, 𝑌 be Banach spaces such that 𝑋 is reflexive, and the inclusion 𝑋⊂𝑌 is continuous. Assume that {𝑢_𝑛} is bounded sequence in 𝐿^∞(𝑡₀,𝑇;𝑋) such that 𝑢_𝑛→𝑢 weakly in 𝐿^𝑞(𝑡₀,𝑇;𝑋) for some 𝑞∈[1,+∞) and 𝑢∈𝐶⁰([𝑡₀,𝑇];𝑌). Then, 𝑢(𝑡)∈𝑋 for all 𝑡∈[𝑡₀,𝑇] and ‖𝑢‖_𝑋≤sup_𝑛≥1‖‖𝑢_𝑛‖‖_{𝐿∞(𝑡0,𝑇;𝑋)},∀𝑡∈𝑡₀,T.(4.1)

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

Theorem 4.2. Assume that 𝑓 satisfies (1.3)-(1.5), 𝑔 satisfies (H 2), (3.18) and the following conditions lim_{𝜏→−∞}𝑒^𝜆1𝜏^𝜏𝜏+1lim_{𝜏→−∞}||𝑔′(𝑠)||²²𝑑𝑠=0,𝑒^𝜆1𝜏||𝑔(𝑠)||²²=0.(4.2) Then 𝒜(𝜏) satisfies lim_{𝜏→−∞}𝑒^𝜆1𝜏sup_{𝑤∈𝒜(𝜏)}‖𝑤‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}+sup_{𝑤∈𝒜(𝜏)}‖𝑤‖^2𝑆_20(Ω)=0.(4.3)

Proof. We differentiate with respect to time in (1.1), then multiply by 𝑢_𝑡, we get12𝑑𝑑𝑟||𝑢^(𝑟)||²²+‖𝑢(𝑟)‖²=≤ℓ−||𝑢_Ω𝑓′(𝑢)𝑢′(𝑟)𝑢′(𝑟)d𝑥+_Ω𝑔′(𝑟)𝑢′(𝑟)𝑑𝑥(𝑟)||²²+12𝑢′(𝑟)²²+12||𝑔′(𝑟)||²².(4.4) Integrating in the last inequality, in particular, we get ||𝑢^(𝑟)||²²≤||𝑢^(𝑠)||²²+(2ℓ+1)^{𝑡𝜏+𝜖/2}||𝑢^(𝜃)||²²𝑑𝜃+^{𝑡𝜏+𝜖/2}||𝑔^(𝜃)||²²𝑑𝜃.(4.5) for all 𝜏+𝜖/2≤s≤𝜏≤𝑡. Now, integrating with respect to 𝑠, between 𝜏+𝜖/2 and 𝑟𝑟−𝜏−2𝜖||𝑢^(𝑟)||²²≤𝑟−𝜏−2𝜖(2𝑙+1)+1^{𝑡𝜏+𝜖/2}+𝑟−𝜏−𝜖||𝑢^(𝜃)||²²𝑑𝜃2^{𝑡𝜏+𝜖/2}||𝑔^(𝜃)||²²𝑑𝜃.(4.6) for all 𝜏+𝜖/2≤𝑟≤𝑡, in paricular, ||𝑢^(𝑟)||²²≤2𝜖⁻¹𝑟−𝜏−2𝜖(2𝑙+1)+1^{𝑡𝜏+𝜖/2}+||𝑢^(𝜃)||²²𝑑𝜃_{𝑡𝜏+𝜖/2}||𝑔^(𝜃)||²²𝑑𝜃.(4.7) for all 𝑟∈[𝜏+𝜖,𝑡].
Multiplying (1.1) by 𝑢 and then integrating on Ω, we get12𝑑𝑑𝑡|𝑢|²²+‖𝑢‖²+_Ω𝑓(𝑢)𝑢𝑑𝑥=_Ω𝑔(𝑡)𝑢𝑑𝑥≤12𝜆₁||𝑔(𝑡)||²²+𝜆¹2|𝑢|²².(4.8) Using hypothesis (H1) and the fact that ‖𝑢‖²≥(1/2)‖𝑢‖²+(𝜆₁/2)|𝑢|²², we have 𝑑𝑑𝑡|𝑢|²²+𝜆₁|𝑢|²²+2𝐶₁|𝑢|^𝑝𝑝−2𝐶₂||Ω||≤1𝜆₁||𝑔(𝑡)||²².(4.9) Integrating (4.9) from 𝜏 to 𝑟∈[𝜏,𝑡], we have ||𝑢(𝑟)||²²+𝜆₁^𝑟𝜏‖𝑢(𝑠)‖²𝑑𝑠+2𝐶₁^𝑟𝜏||𝑢(𝑠)||^𝑝𝑝𝑑𝑠≤||𝑢(𝜏)||²²+1𝜆₁^𝑟𝜏||𝑔(𝑠)||²²𝑑𝑠+2𝐶₂||Ω||(𝑡−𝜏).(4.10) Thus, ||𝑢(𝑟)||²²+^𝑟𝜏‖𝑢(𝑠)‖²𝑑𝑠+^𝑟𝜏||𝑢(𝑠)||^𝑝𝑝𝑑𝑠≤𝐶||𝑢(𝜏)||²²+^𝑟𝜏||𝑔(𝑠)||²²𝑑𝑠+(𝑡−𝜏).(4.11) Multiplying (1.1) by 𝑢_𝑡 then integrating over Ω, we have ||𝑢′(𝑟)||²²+12𝑑𝑑𝑟‖𝑢(𝑟)‖²+2_Ω𝐹(𝑢(𝑥,𝑟))𝑑𝑥≤12||𝑔(𝑟)||²²+12||𝑢′(𝑟)||²².(4.12) Integrating now between 𝑠∈[𝜏,𝑟] and 𝑟≤𝑡, we obtain ^𝑟𝑠||𝑢′(𝜃)||²²𝑑𝜃+‖𝑢(𝑟)‖²+_Ω𝐹(𝑢(𝑥,𝑟))𝑑𝑥≤^𝑟𝑠||𝑔(𝜃)||²²𝑑𝜃+‖𝑢(𝑠)‖²+_Ω𝐹(𝑢(𝑥,𝑠))𝑑𝑥.(4.13) From (4.13) and using hypothesis (H1), we get ^𝑟𝑠||𝑢′(𝜃)||²²𝑑𝜃+‖𝑢(𝑟)‖²+𝛼₁_Ω≤‖𝑢(𝑠)‖₂||𝑢(𝑥,𝑟)||^𝑝𝑑𝑥−𝛼₂||Ω||+^𝑟𝑠||𝑔(𝜃)||²²𝑑𝜃+𝛼₃_Ω||𝑢(𝑥,𝑟)||^𝑝𝑑𝑥+𝛼₄||Ω||.(4.14) Hence ^𝑟𝑠||𝑢′(𝜃)||²²𝑑𝜃+‖𝑢(𝑟)‖²+𝛼₁𝑢(𝑟)^𝑝𝑝≤‖𝑢(𝑠)‖²+^𝑡𝑠||𝑔(𝜃)||²²𝑑𝜃+𝛼₃||𝑢(𝑠)||^𝑝𝑝+𝛼₂+𝛼₄||Ω||.(4.15) Integrating inequality (4.15) with respect to 𝑠 from 𝜏 to 𝑟, we obtain (𝑟−≤𝜏)𝐶‖𝑢(_𝑟𝜏𝑟)‖²+||𝑢(𝑟)||^𝑝𝑝‖𝑢(𝑠)‖²𝑑𝑠+^𝑟𝜏||𝑢(𝑠)||^𝑝𝑝𝑑𝑠+𝐶(𝑡−𝜏)^𝑡𝜏||𝑔(𝑠)||²²𝑑𝑠+𝐶||Ω||(𝑡−𝜏),(4.16) for all 𝑡≥𝜏, 𝑟∈[𝜏,𝑡].
From (4.16) and (4.11), we obtain that ‖𝑢(𝑟)‖²+||𝑢(𝑟)||^𝑝𝑝≤𝐶||𝑢_𝜏||²²+^𝜏𝜏+2||𝑔(𝑠)||²²𝑑𝑠+1,(4.17) for all 𝑟∈[𝜏+1,𝜏+2].
From (4.7), taking 𝑡=𝜏+3 and 𝜖=2 we have||𝑢′(𝑟)||²²≤(4𝑙+3)^𝜏+3𝜏+1||𝑢′(𝜃)||²²𝑑𝜃+^𝜏+3𝜏+1||𝑔′(𝜃)||²²𝑑𝜃(4.18) for all 𝑟∈[𝜏+2,𝜏+3].
Analogously, and if we take 𝑠=𝜏+1 and 𝑟=𝑡=𝜏+3 in inequlity (4.15), then^𝜏+3𝜏+1||𝑢′(𝑠)||²²𝑑𝑠+‖𝑢(𝜏+3)‖²+𝛼₁||𝑢(𝜏+3)||^𝑝𝑝≤‖𝑢(𝜏+1)‖²+^𝜏𝜏+3||𝑔(𝑠)||²²𝑑𝑠+𝛼₃||𝑢(𝜏+1)||^𝑝𝑝+𝛼₂+𝛼₄||Ω||.(4.19) From (4.18) and (4.19), we obtain ||𝑢^(𝑟)||²²≤(4𝑙+3)‖𝑢(𝜏+1)‖²+𝛼₃||𝑢(𝜏+1)||^𝑝𝑝+(4𝑙+3)𝛼₂+𝛼₄||Ω||+^𝜏𝜏+3||𝑔(𝑠)||²²𝑑𝑠+^𝜏+3𝜏+1||𝑔^(𝜃)||²²𝑑𝜃(4.20) for all 𝑟∈[𝜏+2,𝜏+3].
Owing to this inequality and (4.17), we have||𝑢′(𝑟)||²²≤𝐶||𝑢_𝜏||²²+^𝜏𝜏+3||𝑔(𝑠)||²²+||𝑔^(𝑠)||²²𝑑𝑠+1,(4.21) for all 𝑟∈[𝜏+2,𝜏+3].
From (3.23), (3.60) and using Young’s inequality, we have‖𝑢(𝑟)‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}+‖𝑢(𝑟)‖^2𝑆_20(Ω)≤𝐶||𝑢^(𝑟)||²²+‖𝑢(𝑟)‖²+||𝑢(𝑟)||^𝑝𝑝+1+||𝑔(𝑟)||²²,(4.22) for all 𝑟≥𝜏.
From (4.22) and thank to (4.21), we have‖𝑢(𝑟)‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}+‖𝑢(𝑟)‖^2𝑆_20(Ω)≤+𝐶𝐶||𝑢_𝜏‖||²²𝑢(+𝑟)‖^𝜏𝜏+3₂||𝑔(𝑠)||²²+||𝑔′(𝑠)||²²𝑑𝑠+1+||𝑢(𝑟)||^𝑝𝑝+||𝑔(𝑟)||²²(4.23) for all 𝑟∈[𝜏+2,𝜏+3]. From (4.23) and thanks to (4.17), we have ‖𝑢(𝑟)‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}+‖𝑢(𝑟)‖^2𝑆_20(Ω)≤𝐶||𝑢_𝜏||+^𝜏𝜏+3||𝑔(𝑠)||²²+||𝑔′(𝑠)||²²𝑑𝑠+1+sup_{[]𝑟∈𝜏+2,𝜏+3}||𝑔(𝑟)||²²,(4.24) for all 𝑟∈[𝜏+2,𝜏+3]. Now, observe that by Cauchy’s inequality ||𝑔(𝑟)||≥||𝑔(𝜏+2)||+^𝜏+3𝜏+2||𝑔′(𝑠)||²²𝑑𝑠^1/2,(4.25) for all 𝑟∈[𝜏+2,𝜏+3], 𝜏∈ℝ,𝑢_𝜏∈𝐿²(Ω).
Thus, from (4.23), we have‖‖𝑈(𝜏+2,𝜏)𝑢_𝜏‖‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}+‖‖𝑈(𝜏+2,𝜏)𝑢_𝜏‖‖^2𝑆_20(Ω)≤𝐶||𝑢_𝜏||²²+^𝜏𝜏+3||𝑔(𝑠)||²²+||𝑔′(𝑠)||²²𝑑𝑠+1+||𝑔(𝜏+2)||²².(4.26) for all 𝜏∈ℝ, 𝑢_𝜏∈𝐿²(Ω). From this inequlity, and the fact that 𝒜(𝜏)=𝑈(𝜏,𝜏−2)𝐴(𝜏−2), we obtain ‖𝑣‖^{𝐿2𝑝−2}_{2𝑝−2(Ω)}+‖𝑣‖^2𝑆_20(Ω)≤𝐶sup_{𝑤∈𝐴(𝜏−2)}|𝑤|²²+^{𝜏+1𝜏−2}||𝑔(𝑠)||²²+||𝑔′(𝑠)||²²𝑑𝑠+1+||𝑔(𝜏)||²².(4.27) for all 𝑣∈𝐴(𝜏), and any 𝜏∈ℝ. 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.