On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators

Abstract

We study the long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullback 𝒟-attractor in (𝐿²(Ω))^𝑚 for the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullback 𝒟-attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.

1. Introduction

Nonautonomous equations are of great importance and interest as they appear in many applications in the natural sciences. One way of studying the long-time behavior of solutions of such equations is using the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of such dynamical systems (see [1] and references therein). In recent years, the existence of pullback attractors for reaction-diffusion equations has been studied widely (see, e.g., [2–6]). However, to the best of our knowledge, little seems to be known for the asymptotic behavior of solutions of nonautonomous degenerate equations.

One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of the Grushin type [7] 𝐺_𝑠𝑢=Δ_𝑥𝑢+|𝑥|^2𝑠Δ_𝑦𝑢,𝑋=𝑥,𝑦∈Ω⊂ℝ^𝑁1×ℝ^𝑁2,𝑠≥0. (1.1) The global existence and long-time behavior of solutions to semilinear parabolic equations involving the Grushin operator, in both autonomous and nonautonomous cases, have been studied in some recent works [8–10].

In this paper we consider the following nonautonomous semilinear parabolic system:𝜕𝑢𝜕𝑡−𝑎𝐺_𝑠(𝑢)()𝑢()𝑢()𝑢+𝑓=𝑔𝑋,𝑡,𝑋∈Ω,𝑡>𝜏,𝑋,𝑡=0,𝑋∈𝜕Ω,𝑡>𝜏,𝑋,𝜏=𝑢_𝜏(𝑋),𝑋∈Ω, (1.2) where 𝑋=(𝑥,𝑦)∈Ω⊂ℝ^𝑁1×ℝ^𝑁2(𝑁₁,𝑁₂⩾1), 𝑢_𝜏∈(𝐿²(Ω))^𝑚 is given, 𝑢=(𝑢¹,…,𝑢^𝑚) is an unknown vector-function. Here 𝑎∈Mat_𝑚(ℝ), 𝑓(𝑢)=(𝑓¹(𝑢¹,…,𝑢^𝑚),…,𝑓^𝑚(𝑢¹,…,𝑢^𝑚)), and 𝑔(𝑋,𝑡)=(𝑔¹(𝑋,𝑡),…,𝑔^𝑚(𝑋,𝑡)) satisfy the following conditions:

(H1)𝑎∈Mat_𝑚(ℝ) has a positive symmetric part: (1/2)(𝑎+𝑎^∗)⩾𝛽𝐼_𝑚,𝛽>0;(H2)𝑓∶ℝ^𝑚→ℝ^𝑚 is a 𝐶¹-vector function such that: 𝐶₁|𝑢|^𝑝−𝐶₀≤𝑓(𝑢),𝑢=^𝑚_𝑗=1𝑓^𝑗(𝑢)𝑢^𝑗𝑓(𝑢,𝑝≥2,)≤𝐶₂|𝑢|^𝑝−1+1,−𝐶₃|𝑣|²≤𝑓_𝑢(𝑢)𝑣,𝑣=^𝑚𝑚_𝑖=1𝑗=1𝜕𝑓^𝑖𝜕𝑢^𝑗(𝑢)𝑣^𝑗𝑣^𝑖, (1.3)(1.4)(1.5) where 𝐶₀, 𝐶₁, 𝐶₂, and 𝐶₃ are positive constants; (H3)𝑔∈𝑊^1,2_loc(ℝ;(𝐿²(Ω))^𝑚) such that ⁰_−∞𝑒^{𝜆1𝛽𝑠}𝑔(𝑠)²_(𝐿2(Ω))^𝑚𝑑𝑠<+∞,⁰_−∞^𝑠_−∞𝑒^{𝜆1𝛽𝑟}𝑔(𝑟)²_(𝐿2(Ω))^𝑚𝑑𝑟𝑑𝑠<+∞,_0−∞𝑒^{𝜆1𝛽𝑠}𝑔^′(𝑠)²_{(𝐿2(Ω))𝑚}𝑑𝑠<+∞, (1.6) where 𝜆₁ is the first eigenvalue of the operator 𝐺_𝑠 in Ω with the homogeneous Dirichlet boundary condition.

In order to study problem (1.2), we will use the natural energy space 𝕊¹₀(Ω)∶=(𝑆¹₀(Ω))^𝑚 defined as the complete of (𝐶^∞₀(Ω))^𝑚 in the following norm: ‖𝑢‖_𝕊¹_0(Ω)=_Ω∇_𝑥𝑢²+|𝑥|^2𝑠∇_𝑦𝑢²𝑑𝑋^1/2. (1.7) From the results in [11], we know that the embedding 𝕊¹₀(Ω)↪𝕃^𝑝(Ω) is continuous if 1⩽𝑝⩽2^∗_𝑠∶=(2𝑁(𝑠))/(𝑁(𝑠)−2), where 𝑁(𝑠)∶=𝑁₁+(𝑠+1)𝑁₂; moreover, this embedding is compact if 1⩽𝑝<2^∗_𝑠.

Notations
Denote 𝕃^𝑝(Ω)∶=(𝐿^𝑝(Ω))^𝑚, and 𝕊⁻¹(Ω) the dual space of 𝕊¹₀(Ω). For functions 𝑢,𝑣∶ℝ^𝑁→ℝ^𝑚, we set ((∇𝑢,∇𝑣))∶=^𝑚_𝑖=1∇𝑢^𝑖,∇𝑣^𝑖=^𝑚𝑁_𝑖=1𝑘=1𝜕𝑢^𝑖𝜕𝑋_𝑘𝜕𝑣^𝑖𝜕𝑋_𝑘, (1.8) so if 𝑎=(𝑎_𝑖𝑗)^𝑚_𝑖,𝑗=1∈Mat_𝑚(ℝ), then ((𝑎∇𝑢,∇𝑣))=^𝑚_𝑖,𝑗=1𝑎_𝑖𝑗∇𝑢^𝑖,∇𝑣^𝑗=^𝑚𝑁_{𝑖,𝑗=1𝑘=1}𝑎_𝑖𝑗𝜕𝑢^𝑖𝜕𝑋_𝑘𝜕𝑣^𝑗𝜕𝑋_𝑘, (1.9) where (⋅,⋅) denotes the inner product in ℝ^𝑁.

Noting that by assumption (H1), we have =1((𝑎∇𝑢,∇𝑢))2^𝑚_𝑖,𝑗=1𝑎_𝑖𝑗+𝑎_𝑗𝑖∇𝑢^𝑖,∇𝑢^𝑗≥𝛽^𝑚_𝑗=1∇𝑢^𝑗,∇𝑢^𝑗=𝛽^𝑚_𝑗=1∇𝑢^𝑗2. (1.10) Hence_Ω𝑎∇_𝑥𝑢,∇_𝑥𝑢+|𝑥|^2𝑠𝑎∇_𝑦𝑢,∇_𝑦𝑢𝑑𝑋≥𝛽‖𝑢‖²_𝕊10(Ω),_Ω𝑎𝐺_𝑠𝑢,𝐺_𝑠𝑢𝑑𝑋≥𝛽𝐺_𝑠𝑢²_𝕃2(Ω). (1.11)(1.12)

The aim of this paper is to study the long-time behavior of solutions to problem (1.2) by using the theory of pullback 𝒟-attractors. We first prove, under assumptions (H1)–(H3), the existence of a pullback 𝒟-attractor in 𝕃²(Ω) for the process 𝑈(𝑡,𝜏) associated to problem (1.2). Then, with an additional condition that the system has a special gradient structure, namely, 𝑎=𝛽𝐼_𝑚 and there exists a function 𝐹∶ℝ^𝑚→ℝ such that 𝑓(𝑢)=grad_𝑢𝐹(𝑢), we prove the existence of a pullback 𝒟-attractor in the space 𝕊¹₀(Ω)∩𝕃^𝑝(Ω) for the process 𝑈(𝑡,𝜏). Moreover, we prove the boundedness of the pulback 𝒟-attractor in 𝕃^2𝑝−2(Ω) and in 𝕊²₀(Ω), and give estimates of the fractal dimension of the pulback 𝒟-attractor. It is worth noticing that our results, in particular, extend and improve some recent results on the existence of pullback 𝒟-attractors for the reaction-diffusion equations [3–5] and for the Grushin equations [8].

Let us explain the methods used in the paper. We first prove the existence of a family of pullback 𝒟-absorbing sets in 𝕊¹₀(Ω). Thanks to the compactness of the embedding 𝕊¹₀(Ω)↪𝕃²(Ω), we immediately get the existence of a pullback 𝒟-attractor in 𝕃²(Ω). When the system has a special gradient structure, we are able to prove the existence of a pullback 𝒟-attractor in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω). To do this, we follow the general lines of the approach used in [8], with some modifications so that we can improve conditions imposed on the external force 𝑔. In particular, we use the asymptotic a priori estimate method initiated in [12] to testify the pullback asymptotic compactness of the corresponding process. Moreover, in this case we also prove the regularity of the pullback 𝒟-attractor in the spaces 𝕃^2𝑝−2(Ω) and 𝕊¹₀(Ω). Finally, using the recent results in [13], we give an estimate of the fractal dimension of the pullback 𝒟-attractor. It is noticed that we do not impose the restriction on the exponent 𝑝 in (H2) as in [13].

The rest of the paper is organized as follows. In Section 2, for the convenience of the reader, we recall some concepts and results on pullback 𝒟-attractors which we will use. In Section 3, we prove the existence of a pullback 𝒟-attractor in 𝕃²(Ω) in the general case. In Section 4, under the additional assumption that the system has a gradient structure, we prove the regularity and fractal dimension estimates of the pullback 𝒟-attractor.

2. Preliminaries

2.1. Pullback Attractors

For convenience of the reader, we recall in this section some concepts and results on the theory of pullback 𝒟-attractors, which will be used in the paper.

Let 𝑋 be a metric space with metric 𝑑. Denote by ℬ(𝑋) the set of all bounded subsets of 𝑋. For 𝐴,𝐵⊂𝑋, the Hausdorff semidistance between 𝐴 and 𝐵 is defined by dist(𝐴,𝐵)=sup_𝑥∈𝐴inf_𝑦∈𝐵𝑑𝑥,𝑦. (2.1) 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 verifying the norm-to-weak continuity of a process.

Proposition 2.1 (see [14]). Let 𝑋,𝑌 be two Banach spaces, 𝑋^∗,𝑌^∗ be, respectively, their dual spaces. Assume 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 a compact set of 𝑋 to be a bounded set of 𝑋.

Suppose that 𝒟 is a nonempty class of parameterized sets 𝒟={𝐷(𝑡)∶𝑡∈ℝ}⊂ℬ(𝑋).

Definition 2.2. The process {𝑈(𝑡,𝜏)} is said to be pullback 𝒟-asymptotically compact if for any 𝑡∈ℝ, any 𝒟∈𝒟, and any sequence {𝜏_𝑛}_𝑛 with 𝜏_𝑛≤𝑡 for all 𝑛, and 𝜏_𝑛→−∞, any sequence 𝑥_𝑛∈𝐷(𝜏_𝑛), the sequence {𝑈(𝑡,𝜏_𝑛)𝑥_𝑛} is relatively compact in 𝑋.

Definition 2.3. A process {𝑈(𝑡,𝜏)} is called pullback 𝜔-𝒟-limit compact if for any 𝜀>0, any 𝑡∈ℝ, and 𝒟∈𝒟, there exists a 𝜏₀=𝜏₀(𝒟,𝜀,𝑡)≤𝑡 such that 𝛼_𝜏≤𝜏0𝑈(𝑡,𝜏)𝐷(𝜏)≤𝜀, (2.2) where 𝛼 is the Kuratowski measure of noncompactness of 𝐵∈ℬ(𝑋), 𝛼(𝐵)=inf𝛿>0∣𝐵hasafiniteopencoverofsetsofdiameter≤𝛿. (2.3)

Lemma 2.4 (see [3]). A process {𝑈(𝑡,𝜏)} is pullback 𝒟-asymptotically compact if and only if it is 𝜔-𝒟-limit compact.

Definition 2.5. A family of bounded sets ℬ∈𝒟 is called pullback 𝒟-absorbing for the process {𝑈(𝑡,𝜏)} if for any 𝑡∈ℝ and any 𝒟∈𝒟, there exists 𝜏₀=𝜏₀(𝒟,𝑡)≤𝑡 such that _𝜏≤𝜏0𝑈(𝑡,𝜏)𝐷(𝜏)⊂𝐵(𝑡). (2.4)

Definition 2.6. A family 𝒜={𝐴(𝑡)∶𝑡∈ℝ}⊂ℬ(𝑋) is said to be a pullback 𝒟-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), for all 𝑡≥𝜏;(3)𝒜 is pullback 𝒟-attracting, that is, lim_{𝜏→−∞}dist(𝑈(𝑡,𝜏)𝐷(𝜏),𝐴(𝑡))=0, (2.5) for all 𝒟∈𝒟 and all 𝑡∈ℝ;(4)if {𝐶(𝑡)∶𝑡∈ℝ} is another family of closed attracting sets, then 𝐴(𝑡)⊂𝐶(𝑡), for all 𝑡∈ℝ.

Theorem 2.7 (see [3]). Let {𝑈(𝑡,𝜏)} be a norm-to-weak continuous process such that {𝑈(𝑡,𝜏)} is pullback 𝒟-asymptotically compact. If there exists a family of pullback 𝒟-absorbing sets ℬ={𝐵(𝑡)∶𝑡∈ℝ}∈𝒟, then {𝑈(𝑡,𝜏)} has a unique pullback 𝒟-attractor 𝒜={𝐴(𝑡)∶𝑡∈ℝ} and 𝐴(𝑡)=_{𝑠≤𝑡𝜏≤𝑠}𝑈(𝑡,𝜏)𝐵(𝜏). (2.6)

2.2. Fractal Dimension of Pullback Attractors

Given a compact 𝐾⊂𝑋 and 𝜀>0, we denote by 𝑁(𝐾,𝜀) the minimum number of open balls in 𝑋 with radius 𝜀 which are necessary to cover 𝐾.

Definition 2.8. For any nonempty compact set 𝐾⊂𝑋, the fractal dimension of 𝐾 is the number dim_𝑓(𝐾)∶=lim_𝜀→0log𝑁(𝐾,𝜀).log1/𝜀 (2.7)

Definition 2.9. A bounded subset 𝐵₀⊂𝐻 is called a uniformly pullback absorbing set for process 𝑈(𝑡,𝜏) if for every 𝐵⊂𝐻 is bounded, there exists a 𝜏₀≥0 such that 𝑈𝑡,𝑡−𝜏₀𝐵∈𝐵₀,∀𝜏≥𝜏₀, (2.8) here, 𝜏₀ does not depend on the choice of 𝑡.

Theorem 2.10 (see [13]). Let 𝑈(𝑡,𝜏) be a process in a separable Hilbert space 𝐻, 𝐵 be a uniformly pullback absorbing set in 𝐻, 𝐴={𝐴(𝑡)∶𝑡∈ℝ} be a pullback attractor for 𝑈(𝑡,𝜏), if there exists a finite dimensional projection 𝑃 in the space 𝐻 such that 𝑃𝑈𝑡,𝑡−𝑇₀𝑢₁−𝑈(𝑡,𝑡−𝑇₀)𝑢_2𝐻≤𝑙𝑇₀𝑢₁−𝑢_2𝐻 (2.9) for all 𝑢₁,𝑢₂∈𝐵 and some 𝑇₀,𝑙(𝑇₀)>0 and (𝐼−𝑃)𝑈𝑡,𝑡−𝑇₀𝑢₁−𝑈𝑡,𝑡−𝑇₀𝑢_2𝐻≤𝛿𝑢₁−𝑢₂ (2.10) for all 𝑢₁,𝑢₂∈𝐵, where 𝛿<1, 𝑇₀ and 𝑙(𝑇₀) are independent of the choice of 𝑡. Then the family of pullback attractors 𝐴={𝐴(𝑡)∶𝑡∈ℝ} possesses a finite fractal dimension especifically dim_𝑓(𝐴(𝑡𝑇))≤dim𝑃log1+8𝑙₀21−𝛿log1+𝛿⁻¹,∀𝑡∈ℝ. (2.11)

3. Existence of Pullback 𝒟-Attractors in 𝕃²(Ω)

Denote 𝑉∶=𝐿^𝑝𝜏,𝑇;𝕃^𝑝(Ω)∩𝐿²𝜏,𝑇;𝕊¹₀(Ω,𝑉_∗)∶=𝐿²𝜏,𝑇;𝕊⁻¹(Ω)+𝐿^𝑝′𝜏,𝑇;𝕃^𝑝′(Ω), (3.1) where 𝑝^′ is the conjugate of 𝑝 (i.e., 1/𝑝+1/𝑝^′=1).

Definition 3.1. Let 𝑇>0 and 𝑢_𝜏∈𝕃²(Ω) be given. A function 𝑢 is called a weak solution of problem (1.2) on (𝜏,𝑇) if 𝑢∈𝑉,𝜕𝑢𝜕𝑡∈𝑉^∗,𝑢|_𝑡=𝜏=𝑢_𝜏a.e.inΩ,_𝑇𝜏Ω𝑢_𝑡,𝜑+𝑎∇_𝑥𝑢,∇_𝑥𝜑+|𝑥|^2𝑠𝑎∇_𝑦𝑢,∇_𝑦𝜑+𝑓(𝑢)=,𝜑𝑑𝑋𝑑𝑡_𝑇𝜏Ω𝑔(𝑡),𝜑𝑑𝑋𝑑𝑡 (3.2) for all test functions 𝜑∈𝑉.

One can prove that if 𝑢∈𝑉 and 𝜕𝑢/𝜕𝑡∈𝑉^∗, then 𝑢∈𝐶([0,𝑇];𝕃²(Ω)) (see [10]). This makes the initial condition in (1.2) meaningful.

Theorem 3.2. Under assumptions (H1)–(H3), for any 𝜏∈ℝ, 𝑇>𝜏, 𝑢_𝜏∈𝕃²(Ω) given, problem (1.2) has a unique weak solution 𝑢 on (𝜏,𝑇). Moreover, the solution 𝑢 exists on the interval (𝜏,+∞) and the following inequality holds: ‖𝑢‖²_𝕃2(Ω)≤𝑒^{−𝜆1𝛽(𝑡−𝜏)}𝑢_𝜏²_𝕃2(Ω)+2𝐶₀|Ω|𝜆₁𝛽+𝑒^{−𝜆1𝛽𝑡𝑡}_−∞𝑒^{𝜆1𝛽𝑠}𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠,∀𝑡≥𝜏. (3.3)

Proof. The existence and uniqueness of a weak solution to problem (1.2) are proved similarly to the scalar case in [10], so it is omitted here.
We now prove inequality (3.3). Multiplying (1.2) by 𝑢, integrating over Ω, and using (1.11), we have12𝑑‖𝑢‖𝑑𝑡^2𝕃2(Ω)+𝛽‖𝑢‖²_𝕊10(Ω)+_Ω𝑓(𝑢),𝑢𝑑𝑋=_Ω𝑔(𝑡),𝑢𝑑𝑋. (3.4) Using condition (1.3) and the Cauchy inequality, we obtain 𝑑‖𝑢‖𝑑𝑡^2𝕃2(Ω)+2𝛽‖𝑢‖²_𝕊10(Ω)+2𝐶₁‖𝑢‖^𝑝_𝕃𝑝(Ω)−2𝐶₀|Ω|≤2𝛽𝜆₁𝑔(𝑡)²_𝕃2(Ω)+𝛽𝜆₁2‖𝑢‖²_𝕃2(Ω). (3.5) Because ‖𝑢‖²_𝕊10(Ω)≥𝜆₁‖𝑢‖²_𝕃2(Ω), so (3.5) becomes 𝑑‖𝑢‖𝑑𝑡^2𝕃2(Ω)+𝜆₁𝛽‖𝑢‖²_𝕃2(Ω)≤2𝐶₀|Ω|+1𝜆₁𝛽𝑔(𝑡)²_𝕃2(Ω). (3.6) Applying the Gronwall inequality we get (3.3).

Now, we can define the family of two-parameter mappings 𝑈(𝑡,𝜏)∶𝕃²(Ω)⟶𝕊¹₀(Ω)∩𝕃^𝑝(Ω),𝑢_𝜏⟼𝑈(𝑡,𝜏)𝑢_𝜏, (3.7) where 𝑈(𝑡,𝜏)𝑢_𝜏=𝑢(𝑡) is the unique weak solution of (1.2) with the initial datum 𝑢_𝜏 at time 𝜏. Then 𝑈 defines a continuous process on 𝕃²(Ω).

Let ℛ be the set of all functions 𝑟∶ℝ→(0,+∞) such that lim_{𝑡→−∞}𝑒^{𝜆1𝛽𝑡}𝑟²(𝑡)=0 and denote by 𝒟 the class of all families 𝒟={𝐷(𝑡)∶𝑡∈ℝ}⊂ℬ(𝕊¹₀(Ω)) such that 𝐷(𝑡)⊂𝐵(𝑟(𝑡)) for some 𝑟(𝑡)∈ℛ, where 𝐵(𝑟(𝑡)) is the closed ball in 𝕊¹₀(Ω) with radius 𝑟(𝑡).

Lemma 3.3. Under assumptions (H1)–(H3), there exists a constant 𝐶>0 such that the solution 𝑢 of problem (1.2) satisfies the following inequality for all 𝑡>𝜏: ‖𝑢‖²_𝕊10(Ω)()+1≤𝐶1+𝑡−𝜏𝑒𝑡−𝜏^{−𝛼(𝑡−𝜏)}𝑢_𝜏²_𝕃2(Ω)+11++1𝑡−𝜏1+𝑒𝑡−𝜏^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+1𝑑𝑠1+𝑒𝑡−𝜏^{−𝛼𝑡𝑡}_−∞^𝑠_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠, (3.8) where 𝛼=𝛽𝜆₁. This implies that there exists a family of pullback 𝒟-absorbing sets in 𝕊¹₀(Ω) for the process {𝑈(𝑡,𝜏)}.

Proof. We multiply (1.2) by −𝐺_𝑠𝑢 and integrate over Ω. After some standard transformations we obtain 12𝑑‖‖𝑑𝑡𝑢(𝑡)²_𝕊10(Ω)+𝛽𝐺_𝑠𝑢(𝑡)²_𝕃2(Ω)⩽_Ω𝑓(𝑢(𝑡)),Δ_𝑥𝑢(𝑡)+|𝑥|^2𝑠𝑓(𝑢),Δ_𝑦𝑢(𝑡)𝑑𝑋+_Ω𝑔,𝐺_𝑠𝑢(𝑡)𝑑𝑋. (3.9) Without loss of generality, we may assume that 𝑓(0)=0. Otherwise we can replace 𝑓(𝑢) by 𝑓(𝑢)=𝑓(𝑢)−𝑓(0). The function 𝑓 satisfies the same conditions with modified constants 𝐶_𝑖(𝑖=0,1,2,3), because |𝑓(0)|⩽𝐶₂ (see (1.4)). Hence, since 𝑓(𝑢(𝑡))|_𝜕Ω=0, we get _Ω𝑓(𝑢),Δ_𝑥𝑢(𝑡)𝑑𝑋=^𝑁1𝑚_{𝑘=1𝑖=1Ω}𝑓^𝑖(𝑢)𝜕²𝑢^𝑖𝜕𝑋²_𝑘𝑑𝑋=−^𝑁1𝑚_𝑘=1^𝑚_{𝑖=1𝑗=1Ω}𝜕𝑓^𝑖𝜕𝑢^𝑗(𝑢)𝜕𝑢^𝑗𝜕𝑋_𝑘𝜕𝑢^𝑖𝜕𝑋_𝑘𝑑𝑋=−^𝑁1_𝑘=1Ω𝑓_𝑢(𝑢)𝜕𝑢𝜕𝑋_𝑘,𝜕𝑢𝜕𝑋_𝑘𝑑𝑋≤𝐶₃^𝑁1_𝑘=1Ω𝜕𝑢𝜕𝑋_𝑘²𝑑𝑋=𝐶_3Ω∇_𝑥𝑢²𝑑𝑋, (3.10) where we have used condition (1.5). Similarly, we have _Ω|𝑥|^2𝑠𝑓(𝑢),Δ_𝑦𝑢(𝑡)𝑑𝑋≤𝐶_3Ω|𝑥|^2𝑠∇_𝑦𝑢²𝑑𝑋. (3.11) Hence _Ω𝑓(𝑢),Δ_𝑥𝑢(𝑡)𝑑𝑋+_Ω|𝑥|^2𝑠𝑓(𝑢),Δ_𝑦𝑢(𝑡)≤𝐶₃‖𝑢(𝑡)‖²_𝕊10(Ω). (3.12) By the Cauchy inequality we have _Ω𝑔,𝐺_𝑠𝑢(𝑡)1𝑑𝑋≤𝑔2𝛽^2𝕃2(Ω)+𝛽2𝐺_𝑠𝑢(𝑡)²_𝕃2(Ω). (3.13) From (3.9)–(3.13) we obtain 𝑑‖‖𝑑𝑡𝑢(𝑡)²_𝕊10(Ω)+𝛽𝐺_𝑠𝑢(𝑡)²_𝕃2(Ω)≤2𝐶₃‖𝑢(𝑡)‖²_𝕊10(Ω)+1𝛽𝑔²_𝕃2(Ω), (3.14) thus, 𝑑‖‖𝑑𝑡𝑢(𝑡)²_𝕊10(Ω)+𝛼‖𝑢(𝑡)‖²_𝕊10(Ω)⩽2𝐶₃‖𝑢(𝑡)‖²_𝕊10(Ω)+1𝛽𝑔²_𝕃2(Ω), (3.15) where 𝛼=𝛽𝜆₁. Multiplying (3.15) by (𝑡−𝜏)𝑒^𝛼𝑡 and integrating from 𝜏 to 𝑡, we obtain (𝑡−𝜏)𝑒^𝛼𝑡‖𝑢‖²_𝕊10(Ω)≤2𝐶₃(𝑡−𝜏)+1^𝑡_𝜏𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕊10(Ω)𝑑𝑠+𝑡−𝜏𝛽^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (3.16) Multiplying (3.3) by 𝛼𝑒^𝛼𝑡 and integrating from 𝜏 to 𝑡, we have 𝛼^𝑡_𝜏𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕃2(Ω)𝑑𝑠≤𝛼(𝑡−𝜏)𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+2𝐶₀|Ω|𝛼𝑒^𝛼𝑡+^𝑡_−∞^𝑠_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠. (3.17) Now, from (3.5) we get 𝑑‖𝑢‖𝑑𝑡^2𝕃2(Ω)+𝛽‖𝑢‖²_𝕊10(Ω)≤1𝛼𝑔(𝑡)²_𝕃2(Ω)+2𝐶₁|Ω|. (3.18) Multiplying this equation by 𝑒^𝛼𝑡 and integrating from 𝜏 to 𝑡, we deduce that 𝑒^𝛼𝑡‖𝑢(𝑡)‖²_𝕃2(Ω)+𝛽^𝑡_𝜏𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕊10(Ω)𝑑𝑠≤𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+2𝐶₁|Ω|𝛼𝑒^𝛼𝑡+1𝛼^𝑡_𝜏𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠+𝛼^𝑡_𝜏𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕃2(Ω)𝑑𝑠. (3.19) Using (3.17), (3.19) becomes 𝑒^𝛼𝑡‖𝑢(𝑡)‖²_𝕃2(Ω)+𝛽^𝑡_𝜏𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕊10(Ω)𝑑𝑠≤𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝐶₀+𝐶₁|Ω|𝛼𝑒^𝛼𝑡+𝛼(𝑡−𝜏)𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+1𝛼^𝑡_−∞𝑒^𝛼𝜏𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠+^𝑡_−∞^𝑠_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠. (3.20) Substituting (3.20) into (3.16) we obtain ‖𝑢(𝑡)‖²_𝕊10(Ω)≤𝑒^{−𝛼(𝑡−𝜏)}2𝐶₃+1𝜆₁+2𝐶₃𝜆₁()+1𝑡−𝜏𝛽()𝑢𝑡−𝜏^{𝜏2𝕃2(Ω)}+2𝐶₃+1𝐶𝑡−𝜏⁰+𝐶₁|Ω|+1𝛼𝛽𝛼𝛽2𝐶₃1+𝛼+𝑒𝑡−𝜏^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+1𝑑𝑠𝛽2𝐶₃+1𝑒𝑡−𝜏^{−𝛼𝑡𝑡}_−∞^𝑠_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠. (3.21) Hence we get (3.8) with 𝐶=𝐶(𝛽,𝐶₀,𝐶₁,𝐶₃,𝜆₁).
Let 𝑟²₀(𝑡) be the right-hand side of (3.8), and let 𝐵₀(𝑟₀(𝑡)) be the closed ball in 𝕊¹₀(Ω) centered at 0 with radius 𝑟₀(𝑡). Obviously for any 𝒟∈𝒟 and any 𝑡∈ℝ, by (3.8) there exists 𝜏₀=𝜏₀(𝒟)≤𝑡 such that the solution 𝑢 with initial datum 𝑢_𝜏∈𝒟(𝜏) at time 𝜏 satisfies ‖𝑢(𝑡)‖_𝕊¹_0(Ω)≤𝑟₀(𝑡) for all 𝜏≤𝜏₀; that is, ℬ={𝐵₀(𝑟₀(𝑡))∶𝑡∈ℝ} is a family of bounded pullback 𝒟-absorbing sets in 𝕊¹₀(Ω).

From the above lemma we deduce that the process {𝑈(𝑡,𝜏)} maps a compact set of 𝕊¹₀(Ω) to be a bounded set of 𝕊¹₀(Ω), and thus by Proposition 2.1, the process {𝑈(𝑡,𝜏)} is norm-to-weak continuous in 𝕊¹₀(Ω). Since {𝑈(𝑡,𝜏)} has a family of pullback 𝒟-absorbing sets in 𝕊¹₀(Ω) and the embedding 𝕊¹₀(Ω)↪𝕃²(Ω) is compact, we immediately get the following.

Theorem 3.4. Under assumptions (H1)–(H3), the process {𝑈(𝑡,𝜏)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕃²(Ω).

4. Some Further Results in the Gradient Case

In this section, instead of (H1)–(H3), we assume that (H1bis)𝑎=𝛽𝐼_𝑚, where 𝐼_𝑚 is the unit matrix and 𝛽>0; (H2bis)𝑓 satisfies (H2) and 𝑓(𝑢)=grad_𝑢𝐹(𝑢)=((𝜕𝐹/𝜕𝑢¹)(𝑢),…,(𝜕𝐹/𝜕𝑢^𝑚)(𝑢)), where 𝐹∶ℝ^𝑚→ℝ is a potential function satisfying 𝐶₁|𝑢|^𝑝−𝐶₀≤𝐹(𝑢)≤𝐶₂|𝑢|^𝑝+𝐶₀,∀𝑢∈ℝ^𝑚, (4.1) with 𝐶₁, 𝐶₂, 𝐶₀ being positive constants (H3bis)𝑔∈𝑊^1,2_loc(ℝ,𝕃²(Ω)) satisfies ⁰_−∞𝑒^𝛼𝑡𝑔(𝑡)²_𝕃2(Ω)+𝑔^′(𝑡)²_𝕃2(Ω)𝑑𝑡<+∞, (4.2) where 𝛼=𝛽𝜆₁.

The aim of this section is to prove that the pullback 𝒟-attractor obtained in Section 3 is more regular and has a finite fractal dimension.

4.1. Existence of a Pullback 𝒟-Attractor in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω)

Denote by ℛ the set of all functions 𝑟∶ℝ→(0,+∞) such that lim_{𝑡→−∞}𝑒^{𝜆1𝛽𝑡}𝑟²(𝑡)=0 and denote by 𝒟 the class of all families 𝒟={𝐷(𝑡)∶𝑡∈ℝ}⊂ℬ(𝕊¹₀(Ω)∩𝕃^𝑝(Ω)) such that 𝐷(𝑡)⊂𝐵(𝑟(𝑡)) for some 𝑟(𝑡)∈ℛ, where 𝐵(𝑟(𝑡)) is the closed ball in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω) with radius 𝑟(𝑡). Thanks to the above gradient structure, one can prove the existence of a pullback 𝒟-attractor, not only in 𝕃²(Ω), but also in the space 𝕊¹₀(Ω)∩𝕃^𝑝(Ω) for the process {𝑈(𝑡,𝜏)}.

We first prove the following.

Lemma 4.1. Under assumptions (H1bis)–(H3bis), the solution 𝑢 of problem (1.2) satisfies the following inequality for all 𝑡>𝜏: ‖𝑢‖²_𝕊10(Ω)+‖𝑢‖^𝑝_𝕃𝑝(Ω)≤𝐶𝑒^{−𝛼(𝑡−𝜏)}𝑢_𝜏²_𝕃2(Ω)+1+𝑒^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟, (4.3) where 𝐶=𝐶(𝐶₀,𝐶₁,𝐶₁,𝐶₀,𝛽,𝜆₁). This implies that there exists a family of pullback 𝒟-absorbing sets in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω) for the process {𝑈(𝑡,𝜏)}.

Proof. Using (3.5) with 𝛼=𝜆₁𝛽 and the fact that ‖𝑢‖²_𝕊10(Ω)≥𝜆₁‖𝑢‖²_𝕃2(Ω), we have 𝑑‖𝑢‖𝑑𝑡^2𝕃2(Ω)+𝛼‖𝑢‖²_𝕃2(Ω)+𝛽2‖𝑢‖²_𝕊10(Ω)+2𝐶₁‖𝑢‖^𝑝_𝕃𝑝(Ω)≤2𝐶₀|Ω|+2𝛼𝑔(𝑡)²_𝕃2(Ω), (4.4) thus 𝑑𝑒𝑑𝑡^𝛼𝑡‖𝑢‖²_𝕃2(Ω)+𝐶𝑒^𝛼𝑡𝛽‖𝑢‖²_𝕊10(Ω)+2‖𝑢‖^𝑝_𝕃𝑝(Ω)≤2𝐶₀|Ω|𝑒^𝛼𝑡+2𝛼𝑒^𝛼𝑡𝑔(𝑡)²_𝕃2(Ω). (4.5) Integrating from 𝜏 to 𝑠,𝜏≤𝑠≤𝑡+1, and in particular, we have 𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕃2(Ω)≤𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)𝐶+2⁰𝛼|Ω|𝑒^𝛼𝑠+2𝛼^𝑠_𝜏𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟. (4.6) Furthermore, multiplying (4.5) from 𝑠 to 𝑠+1 and using (4.6) we obtain _𝑠^𝑠+1𝑒^𝛼𝑟𝛽‖𝑢(𝑟)‖²_𝕊10(Ω)+2‖𝑢(𝑟)‖^𝑝_𝕃𝑝(Ω)𝑑𝑟≤𝑒^𝛼𝑠‖𝑢(𝑠)‖²_𝕃2(Ω)𝐶+2⁰|Ω|𝛼𝑒^𝛼𝑠+2𝛼_𝑠^𝑠+1𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑒𝑑𝑟≤𝐶^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑠+_𝑠^𝑠+1𝑒^𝛼𝑟‖𝑢(𝑟)‖²_𝕃2(Ω)+^𝑠_𝜏𝑒^𝛼𝑟‖𝑢(𝑟)‖²_𝕃2(Ω)≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_𝜏𝑒^𝛼𝑟‖𝑢(𝑟)‖²_𝕃2(Ω). (4.7) By assumption (H2bis), then (4.7) becomes _𝑠^𝑠+1𝑒^𝛼𝑟𝛽‖𝑢(𝑟)‖²_𝕊10(Ω)+2𝐹(𝑢(𝑟))𝑑𝑟≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_𝜏𝑒^𝛼𝑟‖𝑢(𝑟)‖²_𝕃2(Ω). (4.8) Multiplying (1.2) by 𝜕𝑢/𝜕𝑡 and integrating over Ω, we have 𝑢_𝑡²_𝕃2(Ω)+12𝑑𝛽‖𝑢‖𝑑𝑡^2𝕊10(Ω)+_Ω𝐹(𝑢)𝑑𝑋=_Ω𝑔(𝑡),𝑢_𝑡1𝑑𝑋≤2𝑔(𝑡)²_𝕃2(Ω)+12𝑢_𝑡²_𝕃2(Ω), (4.9) thus 𝑑𝑒𝑑𝑡^𝛼𝑡𝛽‖𝑢(𝑡)‖_𝕊¹_0(Ω)+2_Ω𝐹(𝑢(𝑡))𝑑𝑋≤𝑒^𝛼𝑡𝛽‖𝑢(𝑡)‖_𝕊¹_0(Ω)+2_Ω𝐹(𝑢(𝑡))𝑑𝑋+𝑒^𝛼𝑡𝑔(𝑡)²_𝕃2(Ω). (4.10) Using (4.8), (4.10), and the uniform Gronwall inequality, we get 𝑒^𝛼𝑡𝛽‖𝑢(𝑡)‖²_𝕊10(Ω)+2_Ω𝐹(𝑢(𝑡))𝑑𝑋≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_𝜏𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟. (4.11) Now, using (H2bis) once again we have from (4.11) that 𝑒^𝛼𝑡𝛽‖𝑢(𝑡)‖²_𝕊10(Ω)+2‖𝑢(𝑡)‖^𝑝_𝕃𝑝(Ω)≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_𝜏𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)𝑑𝑟. (4.12) Thus we obtain (4.3) with a suitable positive constant 𝐶=𝐶𝐶₀,𝐶₁,𝐶₁,𝐶₀,𝛽,𝜆₁. (4.13) Hence, by the argument as in the end of the proof of Lemma 3.3, we obtain a family of bounded pullback 𝒟-absorbing sets in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω).

To prove that the process {𝑈(𝑡,𝜏)} is pullback 𝒟-asymptotically compact in 𝕃^𝑝(Ω), we need the following lemma.

Lemma 4.2 (see [8, Lemma 3.6]). Let {𝑈(𝑡,𝜏)} be a norm-to-weak continuous process in 𝕃²(Ω) and 𝕃^𝑝(Ω), and let {𝑈(𝑡,𝜏)} satisfy the following two conditions: (i){𝑈(𝑡,𝜏)} is pullback 𝒟-asymptotically compact in 𝕃²(Ω);(ii)for any 𝜀>0, ℬ∈𝒟, there exist constants 𝑀(𝜀,ℬ) and 𝜏₀(𝜀,ℬ)≤𝑡 such that _{Ω(|𝑈(𝑡,𝜏)𝑢𝜏|≥𝑀)}𝑈(𝑡,𝜏)𝑢_𝜏^𝑝1/𝑝<𝜀,forany𝑢_𝜏∈𝐵(𝜏),𝜏≤𝜏₀. (4.14) Then {𝑈(𝑡,𝜏)} is pullback 𝒟-asymptotically compact in 𝕃^𝑝(Ω).

Theorem 4.3. Under assumptions (H1bis)–(H3bis), the process {𝑈(𝑡,𝜏)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕃^𝑝(Ω).

Proof. It is sufficient to show that the process {𝑈(𝑡,𝜏)} satisfies the condition (ii) in Lemma 4.2. We will give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
Let 𝑀 be a positive number, we will write 𝑢≥𝑀 (or 𝑢≤−𝑀) as any component of 𝑢 is greater than or equal to 𝑀 (or as any component of 𝑢 is less than or equal to −𝑀).
Using (1.3), (1.4), and for 𝑢≥𝑀 large enough, we have𝑓(𝑢),𝑢−𝑀≥𝑓(𝑢),𝑢−𝑀√𝑚𝑓(𝑢≥𝐶₃)|𝑢|^𝑝−𝐶₂|𝑢|^𝑝−1≥𝐶₄2|𝑢−𝑀|^𝑝+𝛼𝑝|𝑢−𝑀|²𝐶,with0<⁴2<𝐶₃, (4.15) because lim_{|𝑢|→+∞}(𝐶₃|𝑢|^𝑝−𝐶₂|𝑢|^𝑝−1)/((𝐶₄/2)|𝑢−𝑀|^𝑝+(𝛼/𝑝)|𝑢−𝑀|²)>1.
Multiplying (1.2) by (𝑢−𝑀)₊|(𝑢−𝑀)₊|^𝑝−2 and integrating over Ω we obtain1𝑝𝑑𝑑𝑡_{Ω(𝑢≥𝑀)}|𝑢−𝑀|^𝑝𝑑𝑋+_{Ω(𝑢≥𝑀)}𝑎∇_𝑥(𝑢−𝑀),∇_𝑥(𝑢−𝑀)|𝑢−𝑀|^𝑝−2+𝑑𝑋_{Ω(𝑢≥𝑀)}|𝑥|^2𝑠𝑎∇_𝑦(𝑢−𝑀),∇_𝑦(𝑢−𝑀)|𝑢−𝑀|^𝑝−2𝑑𝑋+_{Ω(𝑢≥𝑀)}|𝑢−𝑀|^𝑝−2𝑓(𝑢),𝑢−𝑀𝑑𝑋=_Ω𝑔(𝑡),𝑢−𝑀𝑑𝑋, (4.16) where (𝑢−𝑀)₊∶=𝑢−𝑀if𝑢≥𝑀,0,inothercases. (4.17) On the other hand, by the Cauchy inequality, we have 𝑔(𝑡),(𝑢−𝑀)|𝑢−𝑀|^𝑝−2≤|𝑢−𝑀|^𝑝−1𝑔(𝑡≤𝐶)⁴2|𝑢−𝑀|^2𝑝−2+12𝐶₄𝑔², (4.18) which implies that 𝑔(𝑡),𝑢−𝑀|𝑢−𝑀|^𝑝−2𝐶≥−⁴2|𝑢−𝑀|^2𝑝−2−12𝐶₄𝑔². (4.19) Hence, from (4.15) and (4.19), we have |𝑢−𝑀|^𝑝−2𝑓(𝑢)+𝑔(𝑡)≥𝛼,𝑢−𝑀,𝑢−𝑀𝑝|𝑢−𝑀|^𝑝−12𝐶₄𝑔². (4.20) From (4.16), using (4.20) and noting that _{Ω(𝑢≥𝑀)}𝑎∇_𝑥(𝑢−𝑀),∇_𝑥(𝑢−𝑀)|𝑢−𝑀|^𝑝−2+𝑑𝑋_{Ω(𝑢≥𝑀)}|𝑥|^2𝑠𝑎∇_𝑦(𝑢−𝑀),∇_𝑦(𝑢−𝑀)|𝑢−𝑀|^𝑝−2𝑑𝑋≥0,𝑎=𝛽𝐼_𝑚, (4.21) we have 𝑑‖‖𝑑𝑡𝑢−𝑀^𝑝_{𝕃𝑝(Ω(𝑢≥𝑀))}+𝛼‖𝑢−𝑀‖^𝑝_{𝕃𝑝(Ω(𝑢≥𝑀)))}≤𝑝2𝐶₄𝑔²_𝕃2(Ω). (4.22) Now, multiplying the above inequality by (𝑡−𝜏)𝑒^𝛼𝑡 and integrating from 𝜏 to 𝑡, we get (𝑡−𝜏)𝑒^𝛼𝑡‖𝑢−𝑀‖^𝑝_{𝕃𝑝(Ω(𝑢≥𝑀))}≤^𝑡_𝜏𝑒^𝛼𝑠‖𝑢−𝑀‖^𝑝_{𝕃𝑝(Ω(𝑢≥𝑀))}𝑝𝑑𝑠+2𝐶₄(𝑡−𝜏)^𝑡_𝜏𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)≤𝑑𝑠_𝑡𝜏𝑒^𝛼𝑠‖𝑢‖^𝑝_𝕃𝑝(Ω)𝑝𝑑𝑠+2𝐶₄(𝑡−𝜏)^𝑡_𝜏𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.23) Then (𝑢−𝑀)₊^𝑝_𝕃𝑝(Ω)≤1𝑒𝑡−𝜏^{−𝛼𝑡𝑡}_𝜏𝑒^𝛼𝑠‖𝑢‖^𝑝_𝕃𝑝(Ω)𝑑𝑠+𝑝𝑒^−𝛼𝑡2𝐶₄^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.24) On the other hand, integrating (4.5) from 𝜏 to 𝑡, we have ^𝑡_𝜏𝑒^𝛼𝑠‖𝑢(𝑠)‖^𝑝_𝕃𝑝(Ω)𝑑𝑠≤𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)𝐶+2⁰𝛼|Ω|𝑒^𝛼𝑡+1𝛼^𝑡_𝜏𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.25) Therefore, substituting (4.25) into (4.24), we obtain (𝑢−𝑀)₊^𝑝_𝕃𝑝(Ω)𝑒≤𝐶^{−𝛼(𝑡−𝜏)}𝑢𝑡−𝜏^{𝜏2𝕃2(Ω)}+1+1𝑡−𝜏𝑒𝑡−𝜏^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.26) Hence, for any 𝜀>0, there exists 𝑀₁>0 and 𝜏₁<𝑡 such that for any 𝜏<𝜏₁ and any 𝑀≥𝑀₁, we have _{Ω(𝑢(𝑡)≥𝑀)}|𝑢−𝑀|^𝑝𝑑𝑥≤𝜀. (4.27) Repeating the same step above, just taking (𝑢+𝑀)₋ instead of (𝑢−𝑀)₊, we deduce that there exist 𝑀₂>0 and 𝜏₂<𝑡 such that for any 𝜏<𝜏₂ and any 𝑀≥𝑀₂, _{Ω(𝑢(𝑡)≤−𝑀)}|𝑢+𝑀|^𝑝𝑑𝑥≤𝜀, (4.28) where (𝑢+𝑀)₋=𝑢+𝑀,𝑢≤−𝑀,0,inothercases. (4.29) Let 𝑀₀=max{𝑀₁,𝑀₂} and 𝜏₀=min{𝜏₁,𝜏₂}, we obtain (|𝑢_{Ω(|𝑢|≥𝑀)}|−𝑀)^𝑝𝑑𝑥≤𝜀for𝜏≤𝜏₀,𝑀≥𝑀₀. (4.30) So, we have _{Ω(|𝑢|≥2𝑀)}|𝑢|^𝑝𝑑𝑥=_{Ω(|𝑢|≥2𝑀)}((|𝑢|−𝑀)+𝑀)^𝑝𝑑𝑥≤2^𝑝−1_{Ω(|𝑢|≥2𝑀)}(|𝑢|−𝑀)^𝑝𝑑𝑥+_{Ω(|𝑢|≥2𝑀)}𝑀^𝑝𝑑𝑥≤2^𝑝−1_{Ω(|𝑢|≥𝑀)}(|𝑢|−𝑀)^𝑝𝑑𝑥+_{Ω(|𝑢|≥𝑀)}(|𝑢|−𝑀)^𝑝𝑑𝑥≤2^𝑝𝜀. (4.31) This completes the proof.

To prove the existence of a pullback 𝒟-attractor in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω), we need the folowing lemma.

Lemma 4.4. Under assumptions (H1bis)–(H3bis), for any 𝑡∈ℝ and any bounded subset 𝐵⊂𝕃²(Ω), there exists a positive constant 𝑇=𝑇(𝐵,𝑡)≤𝑡 such that 𝑢_𝑡(𝑡)²_𝕃2(Ω)≤𝐶1+𝑒^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑔^′(𝑠)²_𝕃2(Ω)𝑑𝑠, (4.32) for all 𝜏≤𝑇(𝐵,𝑡) and all 𝑢_𝜏∈𝐵, where 𝐶>0 is independent of 𝑡 and 𝐵.

Proof. We give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
Differentiating (1.2) in time and setting 𝑣=𝑢_𝑡, we get𝑣_𝑡−𝑎𝐺_𝑠𝑣+𝑓^′(𝑢)𝑣=𝑔^′(𝑟). (4.33) Multiplying this inequality by 𝑒^𝛼𝑟𝑣 and integrating over Ω and using (1.11), we get 12𝑑𝑒𝑑𝑟^𝛼𝑟‖𝑣‖²_𝕃2(Ω)+𝛽𝑒^𝛼𝑟‖𝑣‖²_𝕊10(Ω)+𝑒^𝛼𝑟_Ω𝑓^′(𝑢)𝛼𝑣,𝑣𝑑𝑋≤2𝑒^𝛼𝑟‖𝑣‖²_𝕃2(Ω)+𝑒^𝛼𝑟_Ω𝑔^′(𝑟),𝑣𝑑𝑋. (4.34) By the Cauchy inequality and using (1.5), we obtain that 𝑑𝑒𝑑𝑟^𝛼𝑟‖𝑣‖²_𝕃2(Ω)≤2𝐶₃+𝛼+1𝑒^𝛼𝑟‖𝑣‖²_𝕃2(Ω)+𝑒^𝛼𝑟𝑔^′(𝑟)²_𝕃2(Ω). (4.35) Let 𝜏≤𝑠≤𝑡−1. Using (3.5), we have 𝑑𝑒𝑑𝑠^𝛼𝑠‖𝑢‖²_𝕃2(Ω)+𝛽‖𝑢‖²_𝕊10(Ω)+2𝐶₁𝑒^𝛼𝑠‖𝑢‖^𝑝_𝕃𝑝(Ω)≤1𝛼𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝐶₀𝑒^𝛼𝑠|Ω|. (4.36) By (H2bis) we then infer from the above inequality that 𝑑𝑒𝑑𝑠^𝛼𝑠‖𝑢‖²_𝕃2(Ω)+𝐶𝛽𝑒^𝛼𝑠‖𝑢‖²_𝕊10(Ω)+2𝑒^𝛼𝑠_Ω𝐹(𝑢)𝑑𝑋≤𝐶𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑒^𝛼𝑠. (4.37) Integrating this inequality from 𝑟 to 𝑟+1, we obtain _𝑟^𝑟+1𝛽𝑒^𝛼𝑠‖𝑢‖²_𝕊10(Ω)+2𝑒^𝛼𝑠_Ω𝐹(𝑢)𝑒𝑑𝑋𝑑𝑠≤𝐶^𝛼𝑟‖𝑢(𝑟)‖²_𝕃2(Ω)+_𝑟^𝑟+1𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑒^𝛼𝑠𝑑𝑠. (4.38) On the other hand, integrating (4.36) from 𝜏 to 𝑡, we obtain 𝑒^𝛼𝑡‖𝑢‖²_𝕃2(Ω)≤𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+1𝛼^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝐶𝑑𝑠+⁰|Ω|𝛼𝑒^𝛼𝑡. (4.39) So, substituting (4.39) into (4.38), we deduce _𝑟^𝑟+1𝛽𝑒^𝛼𝑠‖𝑢‖²_𝕊10(Ω)+2𝑒^𝛼𝑠_Ω𝐹(𝑢)𝑒𝑑𝑋𝑑𝑠≤𝐶^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠+𝑒^𝛼𝑡<∞,∀𝑟∈[𝜏,𝑡−1]. (4.40) Now multiplying (1.2) by 𝑒^𝛼𝑟𝑣 and integrating over Ω, we have 𝑒^𝛼𝑟‖𝑣‖²_𝕃2(Ω)+𝑑𝑑𝑟𝛽𝑒^𝛼𝑟‖𝑢‖²_𝕊10(Ω)+2𝑒^𝛼𝑟_Ω𝐹(𝑢)𝑑𝑋≤𝛼𝛽𝑒^𝛼𝑟‖𝑢‖²_𝕊10(Ω)+2𝑒^𝛼𝑟_Ω𝐹(𝑢)𝑑𝑋+𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω). (4.41) So applying the uniform Gronwall inequality, we get 𝛽𝑒^𝛼𝑟‖𝑢‖²_𝕊10(Ω)+2𝑒^𝛼𝑟_Ω𝐹(𝑢)𝑑𝑋≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.42) Integrating (4.41) from 𝑟 to 𝑟+1 and by (4.40)–(4.42), we have _𝑟^𝑟+1𝑒^𝛼𝑠‖𝑣‖²_𝕃2(Ω)𝑑𝑠≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.43) Therefore, by (4.35), (4.43), using the uniform Gronwall inequality once again, we get 𝑒^𝛼𝑡‖𝑣‖²_𝕃2(Ω)≤𝐶𝑒^𝛼𝜏𝑢_𝜏²_𝕃2(Ω)+𝑒^𝛼𝑡+^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑔^′(𝑠)²_𝕃2(Ω)𝑑𝑠. (4.44) Hence we get (4.32).

Theorem 4.5. Under assumptions (H1bis)–(H3bis), the process {𝑈(𝑡,𝜏)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω).

Proof. By Lemma 4.1, {𝑈(𝑡,𝜏)} has a family of bounded pullback 𝒟-absorbing sets in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω). It remains to show that {𝑈(𝑡,𝜏)} is pullback 𝒟-asymptotically compact in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω), that is, for any 𝑡∈ℝ, any ℬ∈𝒟, and any sequence 𝜏_𝑛→−∞, any sequence 𝑢_𝜏𝑛∈𝐵(𝜏_𝑛), the sequence {𝑈(𝑡,𝜏_𝑛)𝑢_𝜏𝑛} is precompact in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω). Thanks to Theorem 4.3, we need only to show that the sequence {𝑈(𝑡,𝜏_𝑛)𝑢_𝜏𝑛} is precompact in 𝕊¹₀(Ω).
Let 𝑢_𝑛(𝑡)=𝑈(𝑡,𝜏_𝑛)𝑢_𝜏𝑛. By Theorem 3.4, we can assume that {𝑢_𝑛(𝑡)} is a Cauchy sequence in 𝕃²(Ω). We have𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)²_𝕊10(Ω)=−⟨𝐺_𝑠𝑢_𝑛(𝑡)−𝐺_𝑠𝑢_𝑚(𝑡),𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)⟩=−𝑑𝑢_𝑛(𝑡)−𝑑𝑡𝑑𝑢_𝑚(𝑡)𝑑𝑡,𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)−𝑓𝑢_𝑛(𝑡)−𝑓𝑢_𝑚(𝑡),𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡≤𝑑)𝑢𝑑𝑡^𝑛𝑑(𝑡)−𝑢𝑑𝑡^𝑚(𝑡)²_𝕃2(Ω)𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)²_𝕃2(Ω)+𝐶₃𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)²_𝕃2(Ω), (4.45) where we have used condition (1.5). Because {𝑢_𝑛(𝑡)} is a Cauchy sequence in 𝕃²(Ω) and by Lemma 4.4, one gets 𝑢_𝑛(𝑡)−𝑢_𝑚(𝑡)_𝕊10(Ω)⟶0,as𝑚,𝑛⟶∞. (4.46) The proof is complete.

4.2. 𝕃^2𝑝−2(Ω) and 𝕊²₀(Ω)-Boundedness of the Pullback 𝒟-Attractor

First, we prove the existence of a family of pullback 𝒟-absorbing sets for process 𝑈(𝑡,𝜏) in 𝕃^2𝑝−2(Ω).

Proposition 4.6. Under assumptions (H1bis)–(H3bis), then for any 𝑡∈ℝ and any bounded subset 𝐵⊂𝕃²(Ω), there exists a positive constant 𝜏₀=𝜏₀(𝐵,𝑡)≤𝑡 such that ‖𝑢‖_𝕃^2𝑝−2_{2𝑝−2(Ω)}≤𝐶1+𝑔(𝑡)²_𝕃2(Ω)+𝑒^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑔^′(𝑠)²_𝕃2(Ω)𝑑𝑠 (4.47) for all 𝜏≤𝜏₀ and all 𝑢_𝜏∈𝐵, where 𝐶>0 is independent of 𝑡 and 𝐵.

Proof. Multiplying (1.2) by |𝑢|^𝑝−2𝑢 and integrating over Ω we obtain _Ω𝑎∇_𝑥𝑢,∇_𝑥|𝑢|^𝑝−2𝑢+|𝑥|^2𝑠𝑎∇_𝑦𝑢,∇_𝑦|𝑢|^𝑝−2𝑢𝑑𝑋+_Ω𝑓(𝑢),𝑢|𝑢|^𝑝−2𝑑𝑋=−_Ω𝑢_𝑡,|𝑢|^𝑝−2𝑢𝑑𝑋+_Ω𝑔(𝑡),|𝑢|^𝑝−2𝑢𝑑𝑋. (4.48) By the Cauchy inequality, (1.3) and note that _Ω𝑎∇_𝑥𝑢,∇_𝑥|𝑢|^𝑝−2𝑢+|𝑥|^2𝑠𝑎∇_𝑦𝑢,∇_𝑦|𝑢|^𝑝−2𝑢𝑑𝑋≥0;here𝑎=𝛽𝐼_𝑚, (4.49) then we get 𝐶₁‖𝑢‖_𝕃^2𝑝−2_{2𝑝−2(Ω)}≤1𝐶₁𝑢_𝑡²_𝕃2(Ω)+1𝐶₁𝑔(𝑡)²_𝕃2+𝐶₁2‖𝑢‖_𝕃^2𝑝−2_2𝑝−2. (4.50) Hence, by (4.32) we deduce from (4.16) that 𝐶₁2‖𝑢‖_𝕃^2𝑝−2_{2𝑝−2(Ω)}≤1𝐶₁𝐶1+𝑒^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑔^′(𝑠)²_𝕃2(Ω)+1𝑑𝑠𝐶₁𝑔(𝑡)²_𝕃2(Ω). (4.51) Therefore, we get (4.47) and the proof is complete.

And now, we denote by 𝕊²₀(Ω) the closure of (𝐶^∞₀(Ω))^𝑚 in the norm ‖𝑢‖_𝕊²_0(Ω)=_ΩΔ_𝑥𝑢²+|𝑥|^2𝑠Δ_𝑦𝑢|²𝑑𝑋^1/2. (4.52) It is easy to see that 𝕊²₀(Ω) is a Banach space endowed with the above norm. We now prove the 𝕊²₀(Ω)-boundedness of the pullback 𝒟-attractor.

First, we recall a lemma (see [15]) which is necessary for our proof.

Lemma 4.7. Let 𝑋,𝑌 be Banach spaces such that 𝑋 is reflexive, and the inclusion 𝑋⊂𝑌 is continuous. Assume that {𝑢_𝑛} is a bounded sequence in 𝐿^∞(𝜏,𝑇;𝑋) such that 𝑢_𝑛⇀𝑢 weakly in 𝐿^𝑞(𝜏,𝑇;𝑋) for some 𝑞∈[1,+∞) and 𝑢∈𝐶([𝜏,𝑇];𝑌). Then, 𝑢(𝑡)∈𝑋 for all 𝑡∈[𝜏,𝑇] and ‖𝑢(𝑡)‖_𝑋≤sup_𝑛≥1‖𝑢_𝑛‖_𝐿^∞(𝜏,𝑇;𝑋), for all 𝑡∈[𝜏,𝑇].

Let 𝑢_𝑛(𝑡) be the Galerkin approximations of the solution 𝑢(𝑡) of (1.2) then by Lemma 4.7 with noticing that 𝑢_𝑛=𝑈(𝑡,𝜏)𝑢_𝑛𝜏⇀𝑢=𝑈(𝑡,𝜏)𝑢_𝜏 in 𝐿²(𝜏,𝑇;𝕊¹₀(Ω)) and the inclusion 𝕊²₀(Ω)⊂𝕊¹₀(Ω) is continuous, we only need the estimation on 𝑢(𝑡)=𝑈(𝑡,𝜏)𝑢_𝜏.

Theorem 4.8. Under assumptions (H1bis)–(H3bis), the pullback 𝒟-attractor 𝒜={𝐴(𝑡)∶𝑡∈ℝ} in 𝕊¹₀(Ω)∩𝕃^𝑝(Ω) of the process {𝑈(𝑡,𝜏)} is bounded in 𝕊²₀(Ω). More precisely, for any 𝜏<𝑇₁<𝑇₂, the set ⋃_{𝑡∈[𝑇1,𝑇2]}𝐴(𝑡) is a bounded subset of 𝕊²₀(Ω).

Proof. Let us fix a bounded set 𝐵⊂𝕃²(Ω), 𝜏∈ℝ, 𝜀>0, 𝑡>𝜏+𝜀 and 𝑢_𝜏∈𝐵. Multiplying the first equation in (1.2) by 𝐺_𝑠𝑢 and integrating over Ω, we have _Ω𝑎𝐺_𝑠𝑢(𝑟),𝐺_𝑠𝑢(𝑟)𝑑𝑋=_Ω𝑢^′_𝑟,𝐺_𝑠𝑢(𝑟)𝑑𝑋+_Ω𝑓(𝑢(𝑟)),𝐺_𝑠𝑢(𝑟)−𝑑𝑋_Ω𝑔(𝑟),𝐺_𝑠𝑢(𝑟)𝑑𝑋. (4.53) By the Cauchy inequality we have −_Ω𝑔(𝑟),𝐺_𝑠𝑢(𝑟)2𝑑𝑋≤𝛽𝑔(𝑟)²_𝕃2(Ω)+𝛽8𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω),_Ω𝑢^′_𝑟,𝐺_𝑠𝑢(𝑟)2𝑑𝑋≤𝛽𝑢^′_𝑟²_𝕊10(Ω)+𝛽8𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω). (4.54) Using (3.12), (1.12), and (4.54), then from (4.53) we get 𝛽𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω)≤𝐶₃‖𝑢(𝑟)‖²_𝕊10(Ω)+2𝛽𝑢^′_𝑟²_𝕃2(Ω)+𝑔(𝑟)²_𝕃2(Ω)+𝛽4𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω)=𝐶_3Ω𝑢(𝑟),−𝐺_𝑠𝑢(𝑟)2𝑑𝑋+𝛽𝑢^′_𝑟²_𝕃2(Ω)+𝑔(𝑟)²_𝕃2(Ω)+𝛽4𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω)≤2𝐶²₃𝛽‖𝑢(𝑟)‖²_𝕃2(Ω)+2𝛽𝑢^′_𝑟²_𝕃2(Ω)+2𝛽𝑔(𝑟)²_𝕃2(Ω)+𝛽2𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω). (4.55) Hence, 𝛽2𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω)≤2𝐶²₃𝛽‖𝑢(𝑟)‖²_𝕃2(Ω)+2𝛽𝑢^′_𝑟²_𝕃2(Ω)+2𝛽𝑔(𝑟)²_𝕃2(Ω). (4.56) Differentiating the first equation in (1.2) in time 𝑡 and setting 𝑣(𝑟)=𝑢^′(𝑟), then multiplying by 𝑣(𝑟) and using (1.11) we get 12𝑑‖‖𝑑𝑟𝑣(𝑟)²_𝕃2(Ω)+𝛽‖𝑣(𝑟)‖²_𝕊10(Ω)≤−_Ω𝑓^′_𝑢(𝑢)𝑣(𝑟),𝑣(𝑟)𝑑𝑋+_Ω𝑔^′(𝑟),𝑣(𝑟)𝑑𝑋≤𝐶₃‖𝑣(𝑟)‖²_𝕃2(Ω)+12‖𝑣(𝑟)‖²_𝕃2(Ω)+12𝑔^′(𝑟)²_𝕃2(Ω). (4.57) Hence, 𝑑‖‖𝑑𝑟𝑣(𝑟)²_𝕃2(Ω)≤2𝐶₃+1‖𝑣(𝑟)‖²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω). (4.58) Integrating the above inequality, we have ‖𝑣(𝑟)‖²_𝕃2(Ω)≤‖𝑣(𝑠)‖²_𝕃2(Ω)+2𝐶₃+1^𝑡_𝜏+𝜀/2‖𝑣(𝜃)‖²_𝕃2(Ω)+^𝑡_𝜏+𝜀/2𝑔^′(𝜃)²_𝕃2(Ω)𝑑𝜃, (4.59) for all 𝜏+𝜀/2≤𝑠≤𝑟≤𝑡.
Now, integrating with respect to 𝑠 between 𝜏+𝜀/2 and 𝑟, we get𝜀𝑟−𝜏−2‖𝑣(𝑟)‖²_𝕃2(Ω)≤^𝑟_𝜏+𝜀/2‖𝑣(𝑠)‖²_𝕃2(Ω)+𝑑𝑠2𝐶₃𝜀+1𝑟−𝜏−2^𝑡_𝜏+𝜀/2‖𝑣(𝜃)‖²_𝕃2(Ω)𝜀𝑑𝜃+𝑟−𝜏−2^𝑡_𝜏+𝜀/2𝑔^′(𝜃)²_𝕃2(Ω)≤𝑑𝜃2𝐶₃𝜀+1𝑡−𝜏−2+1^𝑡_𝜏+𝜀/2‖𝑣(𝜃)‖²_𝕃2(Ω)𝜀𝑑𝜃+𝑟−𝜏−2^𝑡_𝜏+𝜀/2𝑔^′(𝜃)²_𝕃2(Ω)𝑑𝜃, (4.60) for all 𝜏+𝜀/2≤𝑟≤𝑡, and in particular, for all 𝑟∈[𝜏+𝜀,𝑡] we have that (from the above estimate) ‖𝑣(𝑟)‖²_𝕃2(Ω)≤2𝜀2𝐶₃𝜀+1𝑡−𝜏−2+1^𝑡_𝜏+𝜀/2‖𝑣(𝜃)‖²_𝕃2(Ω)𝑑𝜃+^𝑡_𝜏𝑔^′(𝜃)²_𝕃2(Ω)𝑑𝜃. (4.61) On the other hand, multiplying the first equation in (1.2) by 𝑣(𝑟) and integrating over Ω, we deduce that ‖𝑣(𝑟)‖²_𝕃2(Ω)+𝛽2𝑑‖‖𝑑𝑟𝑢(𝑟)²_𝕊10(Ω)+_Ω𝑓(𝑢,𝑣)𝑑𝑋≤_Ω𝑔,𝑣𝑑𝑋, (4.62) where we have used (1.11). Using the Cauchy inequality and condition (H2bis), then (4.62) becomes ‖𝑣(𝑟)‖²_𝕃2(Ω)+𝑑𝛽‖𝑢(𝑟𝑑𝑟)‖²_𝕊10(Ω)+2_Ω𝐹(𝑢(𝑟))𝑑𝑋≤𝑔(𝑟)²_𝕃2(Ω). (4.63) Integrating from 𝜏+𝜀/2 to 𝑡 we have ^𝑡_𝜏+𝜀/2‖𝑣(𝜃)‖²_𝕃2(Ω)𝑑𝜃+𝛽‖𝑢(𝑡)‖²_𝕊10(Ω)+2_Ω𝐹(𝑢(𝑡𝑢𝜀))𝑑𝑋≤𝛽𝜏+2²_𝕊¹_0(Ω)+2_Ω𝐹𝑢𝜀𝜏+2𝑑𝑋+^𝑡_𝜏+𝜀/2𝑔(𝜃)²_𝕃2(Ω), (4.64) and hence because of (4.1), we get ^𝑡_𝜏+𝜀/2‖𝑣(𝜃)‖²_𝕃2(Ω)𝑢𝜀𝑑𝜃≤𝛽𝜏+2²_𝕊¹_0(Ω)+2𝐶₂𝑢𝜀𝜏+2^𝑝_𝕃^𝑝_(Ω)+4𝐶₀|Ω|+^𝑡_𝜏𝑔(𝜃)²_𝕃2(Ω). (4.65) Now, substituting (4.65) into (4.61) we deduce ‖𝑣(𝜃)‖²_𝕃2(Ω)≤2𝜀2𝐶₃𝜀+1𝑡−𝜏−2×𝛽𝑢𝜀+1𝜏+2²_𝕊¹_0(Ω)+2𝐶₂𝑢𝜀𝜏+2²_𝕃^𝑝_(Ω)+4𝐶₀|Ω|+^𝑡_𝜏𝑔(𝜃)²_𝕃2(Ω)+𝑑𝜃_𝑡𝜏𝑔^′(𝜃)²_𝕃2(Ω)𝑑𝜃, (4.66) for all 𝑟∈[𝜏+𝜀,𝑡]. Finally, from (4.66) and (4.56) we obtain 𝐺_𝑠𝑢(𝑟)²_𝕃2(Ω)≤8𝛽²𝜀2𝐶₃𝜀+1𝑡−𝜏−2×𝛽𝑢𝜀+1𝜏+2²_𝕊¹_0(Ω)+2𝐶₂𝑢𝜀𝜏+2^𝑝_𝕃^𝑝_(Ω)+4𝐶₀|Ω|+^𝑡_𝜏𝑔(𝜃)²_𝕃2(Ω)+2𝑑𝜃𝛽^𝑡_𝜏𝑔^′(𝜃)²_𝕃2(Ω)2𝑑𝜃+𝛽²𝐶²₃‖𝑢(𝑟)‖²_𝕃2(Ω)+2𝑔(𝑟)²_𝕃2(Ω),∀𝑟∈[𝜏+𝜀,𝑡]. (4.67) Because ‖𝑢‖²_𝕊20(Ω)=‖𝐺_𝑠𝑢‖²_𝕃2(Ω) then from (4.67), the proof is complete.

4.3. Fractal Dimensional Estimates of the Pullback 𝒟-Attractor

Theorem 4.9. Under assumptions (H1bis)–(H3bis), the process 𝑈(𝑡,𝜏) possesses a pullback 𝒟-attractor 𝒜_𝕃²_(Ω) which has a finite fractal dimension in 𝕃²(Ω) and dim_𝑓(𝐴(𝑡))≤𝑘log1+8⋅𝑒^2𝐶321−𝛿log1+𝛿⁻¹,∀𝑡∈ℝ, (4.68) where 𝛿<1,𝑘∈ℕ, and 𝐶₃ in (1.5).

Proof. Let 𝐻_𝑘=span{𝑒₁,𝑒₂,…,𝑒_𝑘}⊂𝕃²(Ω) and 𝑃_𝑘∶𝕃²(Ω)→𝐻_𝑘 be the orthogonal projection, where 𝑒₁,𝑒₂,…,𝑒_𝑗,… are the eigenvectors of the operator −𝐺_𝑠 corresponding to eigenvalues {𝜆_𝑗}^∞_𝑗=1 such that 0<𝜆₁<𝜆₂≤𝜆₃≤⋯≤𝜆_𝑗≤⋯ and 𝜆_𝑗→+∞ as 𝑗→+∞.
From (4.3), we can easily show that there exists a uniformly pullback absorbing set ℬ of process 𝑈(𝑡,𝜏) in 𝕊¹₀(Ω). We set 𝑢₁(𝑡)=𝑈(𝑡,𝜏)𝑢_1𝜏 and 𝑢₂(𝑡)=𝑈(𝑡,𝜏)𝑢_2𝜏 to be solutions associated to problem (1.2) with initial datum 𝑢_1𝜏,𝑢_2𝜏∈ℬ.
Let 𝑤=𝑢₁−𝑢₂, because 𝑢₁, 𝑢₂ being two solutions of (1.2) then we have𝜕𝑤𝜕𝑡−𝑎𝐺_𝑠𝑤+𝑓𝑢₁−𝑓𝑢₂=0. (4.69) Multiplying (4.69) with 𝑤 and integrating over Ω then we have 12𝑑‖‖𝑑𝑡𝑤(𝑡)²_𝕃2(Ω)+𝛽‖𝑤‖²_𝕊10(Ω)+_Ω𝑓𝑢₁−𝑓𝑢₂,𝑤𝑑𝑋≤0, (4.70) here, we have used (1.11).
Using (1.5) then we have𝑑‖‖𝑑𝑡𝑤(𝑡)²_𝕃2(Ω)≤2𝐶₃‖𝑤(𝑡)‖²_𝕃2(Ω). (4.71) Thus, ‖𝑤(𝑡)‖²_𝕃2(Ω)≤𝑒^{2𝐶3(𝑡−𝜏)}‖𝑤(𝜏)‖²_𝕃2(Ω). (4.72) Let 𝑤(𝑡)=𝑤₁(𝑡)+𝑤₂(𝑡) where 𝑤₁(𝑡)∶=𝑃_𝑘𝑤(𝑡) and 𝑤₂(𝑡)∶=(𝐼−𝑃_𝑘)𝑤(𝑡). Therefore, by (4.72) we have 𝑤₁(𝑡)²_𝕃2(Ω)≤𝑒^{2𝐶3(𝑡−𝜏)}‖𝑤(𝜏)‖²_𝕃2(Ω). (4.73) Now, taking the inner product of (4.69) with 𝑤₂ in 𝕃²(Ω), we have 12𝑑𝑤𝑑𝑡²(𝑡)²_𝕃2(Ω)+𝛽𝑤₂²_𝕊10(Ω)≤−𝑓𝑢₁−𝑓𝑢₂,𝑤₂𝑑𝑋. (4.74) Using the Hölder inequality and (1.4), we have −_Ω𝑓𝑢₁−𝑓𝑢₂,𝑤₂𝑑𝑋≤_Ω𝑓𝑢₁−𝑢𝑢₂𝑤₂≤𝑑𝑋_Ω𝑓𝑢₁−𝑓𝑢₂²𝑑𝑋^1/2_Ω𝑤₂²𝑑𝑋^1/2≤𝐶_Ω1+𝑢₁^2𝑝−2+𝑢₂^2𝑝−2𝑑𝑋^1/2𝑤_2𝕃2(Ω)≤𝐶1+𝑢_1𝕃^2𝑝−2_{2𝑝−2(Ω)}+𝑢_2𝕃^2𝑝−2_{2𝑝−2(Ω)}^1/2‖𝑤‖_𝕃²_(Ω)≤𝐶1+𝑢_1𝕃^2𝑝−2_{2𝑝−2(Ω)}+𝑢_2𝕃^2𝑝−2_{2𝑝−2(Ω)}‖𝑤‖_𝕃²_(Ω). (4.75) Therefore, by (4.74), (4.75), and Proposition 4.6 we obtain 12𝑑𝑤𝑑𝑡²(𝑡)²_𝕃2(Ω)+𝛽𝑤₂²_𝕊10(Ω)≤𝐶1+𝑔(𝑡)²_𝕃2(Ω)+𝑒^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑔^′(𝑠)²_𝕃2(Ω)𝑑𝑠‖𝑤(𝑡)‖_𝕃²_(Ω). (4.76) Because ‖𝑤₂(𝑡)‖²_𝕊10(Ω)≥𝜆_𝑘‖𝑤₂(𝑡)‖²_𝕃2(Ω), then (4.76) implies that 𝑑𝑤𝑑𝑡²(𝑡)²_𝕃2(Ω)+2𝛽𝜆_𝑘𝑤₂²_𝕃2(Ω)≤2𝐶1+𝑔(𝑡)²_𝕃2(Ω)+𝑒^{−𝛼𝑡𝑡}_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)+𝑔^′(𝑠)²_𝕃2(Ω)𝑑𝑠‖𝑤(𝑡)‖_𝕃²_(Ω). (4.77) Now, multiplying (4.77) by 𝑒^{𝛽𝜆𝑘𝑡} and integrating from 𝜏 to 𝑡, we get 𝑤₂(𝑡)²_𝕃2(Ω)≤𝑒^{−𝛽𝜆𝑘(𝑡−𝜏)}‖𝑤(𝜏)‖²_𝕃2(Ω)+2𝐶𝑒^{−𝛽𝜆𝑘𝑡𝑡}_𝜏𝑒^{−𝛽𝜆𝑘𝑠}×1+𝑔(𝑠)²_𝕃2(Ω)+𝑒^{−𝛼𝑡𝑠}_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)×‖‖𝑑𝑟𝑤(𝑠)_𝕃²_(Ω)𝑑𝑠. (4.78) Using (4.73) we have 𝑤₂(𝑡)²_𝕃2(Ω)≤𝑒^{−𝛽𝜆𝑘(𝑡−𝜏)}‖𝑤(𝜏)‖²_𝕃2(Ω)+𝐶‖𝑤(𝜏)‖_𝕃²_(Ω)𝑒^{−𝛽𝜆𝑘𝑡𝑡}_𝜏𝑒^{𝛽𝜆𝑘𝑠}𝑒^{𝐶3(𝑠−𝜏)}×1+𝑔(𝑠)²_𝕃2(Ω)+𝑒^{−𝛼𝑠𝑠}_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠≤𝑒^{−𝛽𝜆𝑘(𝑡−𝜏)}‖𝑤(𝜏)‖²_𝕃2(Ω)+𝐶‖𝑤(𝜏)‖_𝕃²_(Ω)𝑒^{𝐶3(𝑡−𝜏)𝑡}_−∞𝑒^{−𝛽𝜆𝑘(𝑡−𝑠)}×1+𝑔(𝑠)²_𝕃2(Ω)+𝑒^{−𝛼𝑠𝑠}_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠≤𝑒^{−𝛽𝜆𝑘(𝑡−𝜏)}‖𝑤(𝜏)‖²_𝕃2(Ω)+𝐶‖𝑤(𝜏)‖_𝕃²_(Ω)𝑒^{𝐶3(𝑡−𝜏)}×1𝛽𝜆_𝑘+^𝑡_−∞𝑒^{−𝛽𝜆𝑘(𝑡−𝑠)}𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠+^𝑡_−∞𝑒^{−𝛽𝜆𝑘(𝑡−𝑠)}𝑒^−𝛼𝑠×^𝑠_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠. (4.79) Now, because ^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠<+∞, (4.80) we can see that, for all 𝑡∈ℝ (see, e.g., [6, Lemma 3.6]), ^𝑡_−∞𝑒^{−𝛽𝜆𝑘(𝑡−𝑠)}𝑔(𝑠)²_𝕃2(Ω)𝑑𝑠⟶0as𝑘⟶+∞, (4.81) and we have ^𝑡_−∞𝑒^{−𝛽𝜆𝑘(𝑡−𝑠)}𝑒^{−𝛽𝜆1𝑠𝑠}_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)≤𝑑𝑟𝑑𝑠_𝑡−∞𝑒^{−𝛽𝜆𝑘𝑡+𝛽(𝜆𝑘−𝜆1)𝑠}𝑑𝑠^𝑡_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)=𝑒𝑑𝑟_−𝛼𝑡𝛽𝜆_𝑘−𝛼^𝑡_−∞𝑒^𝛼𝑠𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)𝑑𝑟. (4.82) Thus, for any 𝑡∈ℝ, from (4.82) we have ^𝑡_−∞𝑒^{−𝛽𝜆𝑘(𝑡−𝑠)}𝑒^{−𝛽𝜆1𝑠𝑠}_−∞𝑒^𝛼𝑟𝑔(𝑟)²_𝕃2(Ω)+𝑔^′(𝑟)²_𝕃2(Ω)𝑑𝑟𝑑𝑠⟶0as𝑘⟶+∞. (4.83) Combining (4.81), (4.83) and taking 𝑇₀=𝑡−𝜏=1, we get 𝑘 is sufficient large then from (4.79) we deduce 𝑤₂(𝑡)²_𝕃2(Ω)≤𝛿‖𝑤(𝜏)‖²_𝕃2(Ω),here0<𝛿<1. (4.84) From (4.73) and (4.31), we have 𝑤₁(𝑡)²_𝕃2(Ω)≤𝑙₀‖𝑤(𝜏)‖²_𝕃2(Ω),𝑤₂(𝑡)²_𝕃2(Ω)≤𝛿‖𝑤(𝜏)‖²_𝕃2(Ω),∀𝑡∈ℝ. (4.85) Here, 𝑙₀=𝑒^2𝐶3; 0<𝛿<1; 𝑇₀=1. Thus, the process 𝑈(𝑡,𝜏) associated to (1.2) satisfies all conditions of Theorem 2.10. This completes the proof.

Acknowledgment

This paper was supported by Vietnam's National Foundation for Science and Technology Development (NAFOSTED), project 101.01-2010.05.