Strong Uniform Attractors for Nonautonomous Suspension Bridge-Type Equations

Abstract

We discuss long-term dynamical behavior of the solutions for the nonautonomous suspension bridge-type equation in the strong Hilbert space 𝐷(𝐴)×𝐻²(Ω)∩𝐻¹₀(Ω), where the nonlinearity 𝑔(𝑢,𝑡) is translation compact and the time-dependent external forces ℎ(𝑥,𝑡) only satisfy condition (𝐶^∗) instead of translation compact. The existence of strong solutions and strong uniform attractors is investigated using a new process scheme. Since the solutions of the nonautonomous suspension bridge-type equation have no higher regularity and the process associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.

1. Introduction

Consider the following equations: 𝑢_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑥}+𝛿𝑢_𝑡+𝑘𝑢⁺()()𝑢()()=𝑙+𝜖ℎ𝑥,𝑡,in0,𝐿×ℝ,0,𝑡=𝑢𝐿,𝑡=𝑢_𝑥𝑥(0,𝑡)=𝑢_𝑥𝑥(𝐿,𝑡)=0,𝑡∈ℝ. (1.1) Suspension bridge equations (1.1) have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and McKenna in 1990. This model has been derived as follows. In the suspension bridge system, suspension bridge can be considered as an elastic and unloaded beam with hinged ends. 𝑢(𝑥,𝑡) denotes the deflection in the downward direction; 𝛿𝑢_𝑡 represents the viscous damping. The restoring force can be modeled owing to the cable with one-sided Hooke’s law so that it strongly resists expansion but does not resist compression. The simplest function to model the restoring force of the stays in the suspension bridge can be denoted by a constant 𝑘 times 𝑢, the expansion, if 𝑢 is positive, but zero, if 𝑢 is negative, corresponding to compression; that is, 𝑘𝑢⁺, where 𝑢⁺=𝑢,if𝑢>0,0,if𝑢⩽0. (1.2) Besides, the right-hand side of (1.1) also contains two terms: the large positive term 𝑙 corresponding to gravity, and a small oscillatory forcing term 𝜖ℎ(𝑥,𝑡) possibly aerodynamic in origin, where 𝜖 is small.

There are many results for (1.1) (cf. [1–9]), for instance, the existence, multiplicity, and properties of the traveling wave solutions, and so forth.

In the study of equations of mathematical physics, attractor is a proper mathematical concept about the depiction of the behavior of the solutions of these equations when time is large or tends to infinity, which describes all the possible limits of solutions. In the past two decades, many authors have proved the existence of attractor and discussed its properties for various mathematical physics models (e.g., see [10–12] and the reference therein). About the long-time behavior of suspension bridge-type equations, for the autonomous case, in [13, 14] the authors have discussed long-time behavior of the solutions of the problem on ℝ² and obtained the existence of global attractors in the space 𝐻²₀(Ω)×𝐿²(Ω) and 𝐷(𝐴)×𝐻²₀(Ω).

It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding nonautonomous model is more natural and precise than the autonomous one, moreover, it always presents as a nonlinear equation, not just a linear one. Therefore, in this paper, we will discuss the following nonautonomous suspension bridge-type equation: let Ω be an open bounded subset of ℝ² with smooth boundary, ℝ_𝜏=[𝜏,+∞], and we add the nonlinear forcing term 𝑔(𝑢,𝑡) (which is dependent on deflection 𝑢 and time 𝑡) to (1.1) and neglect gravity, then we can obtain the following initial-boundary value problem: 𝑢_𝑡𝑡+Δ²𝑢+𝛼𝑢_𝑡+𝑘𝑢⁺+𝑔(𝑢,𝑡)=ℎ(𝑥,𝑡),inΩ×ℝ_𝜏,𝑢(𝑥,𝑡)=Δ𝑢(𝑥,𝑡)=0,on𝜕Ω×ℝ_𝜏,𝑢(𝑥,𝜏)=𝑢₁(𝑥),𝑢_𝑡(𝑥,𝜏)=𝑢₂(𝑥),𝑥∈Ω, (1.3) where 𝑢(𝑥,𝑡) is an unknown function, which could represent the deflection of the road bed in the vertical plane; ℎ(𝑥,𝑡) and 𝑔(𝑢,𝑡) are time dependant external forces; 𝑘𝑢⁺ represents the restoring force, 𝑘 denotes the spring constant; 𝛼𝑢_𝑡 represents the viscous damping, 𝛼 is a given positive constant.

To our knowledge, this is the first time to consider the nonautonomous dynamics of (1.3) with the time dependant external forces ℎ(𝑥,𝑡) and 𝑔(𝑢,𝑡) in the strong topological space 𝐷(𝐴)×𝐻²(Ω)∩𝐻¹₀(Ω). At the same time, in mathematics, we only assume that the force term ℎ(𝑥,𝑡) satisfies the so-called condition (𝐶^∗) (introduced in [15]), which is weaker than translation compact assumption (see [10] or Section 2 below).

This paper is organized as follows. At first, in Section 2, we give (recall) some preliminaries, including the notation we will use, the assumption on nonlinearity 𝑔(⋅,𝑡), and some general abstract results about nonautonomous dynamical system. Then, in Section 3 we prove our main result about the existence of strong attractor for the nonautonomous dynamical system generated by the solution of (1.3).

2. Notation and Preliminaries

With the usual notation, we introduce the spaces 𝐻=𝐿²(Ω),  𝑉=𝐻²(Ω)∩𝐻¹₀(Ω), 𝐷(𝐴)={𝑢∈𝐻²(Ω)∩𝐻¹₀(Ω)∣𝐴𝑢∈𝐿²(Ω)}, where 𝐴=Δ². We equip these spaces with inner product and norm ⟨⋅,⋅⟩,‖⋅‖, ⟨⋅,⋅⟩₁,‖⋅‖₁ and ⟨⋅,⋅⟩₂,‖⋅‖₂, respectively, ⟨𝑢,𝑣⟩=_Ω𝑢(𝑥)𝑣(𝑥)𝑑𝑥,‖𝑢‖²=_Ω|𝑢(𝑥)|²⟨⟩𝑑𝑥,∀𝑢,𝑣∈𝐻,𝑢,𝑣₁=_ΩΔ𝑢(𝑥)Δ𝑣(𝑥)𝑑𝑥,‖𝑢‖²₁=_Ω|Δ𝑢(𝑥)|²⟨⟩𝑑𝑥,∀𝑢,𝑣∈𝑉,𝑢,𝑣₂=_ΩΔ²𝑢(𝑥)Δ²𝑣(𝑥)𝑑𝑥,‖𝑢‖²₂=_ΩΔ²𝑢(𝑥)²𝑑𝑥,∀𝑢,𝑣∈𝐷(𝐴). (2.1) Obviously, we have 𝐷(𝐴)⊂𝑉⊂𝐻=𝐻^∗⊂𝑉^∗, (2.2) where 𝐻^∗,𝑉^∗ is dual space of 𝐻,𝑉, respectively, the injections are continuous, and each space is dense in the following one.

In the following, the assumption on the nonlinearity 𝑔 is given. Let 𝑔 be a 𝐶¹ function from ℝ×ℝ to ℝ and satisfy liminf_|𝑢|→∞𝐺(𝑢,𝑠)𝑢²⩾0, (2.3) where 𝐺(𝑢,𝑠)=∫^𝑢₀𝑔(𝑤,𝑠)𝑑𝑤, and there exists 𝐶₀>0, such that liminf_|𝑢|→∞𝑢,𝑔(𝑢,𝑠)−𝐶₀𝐺(𝑢,𝑠)𝑢²⩾0. (2.4)

Suppose that 𝛾 is an arbitrary positive constant, and 𝑔_𝑢(𝑢,𝑠)⩽𝐶₁1+|𝑢|^𝛾,𝑔_𝑠(𝑢,𝑠)⩽𝐶₁1+|𝑢|^𝛾+1𝐺_𝑠,(𝑢,𝑠)⩽𝛿²𝐺(𝑢,𝑠)+𝐶₂,∀(𝑢,𝑠)∈ℝ×ℝ, (2.5)(2.6) where 𝛿 is a sufficiently small constant.

As a consequence of (2.3)-(2.4), if we denote 𝒢(𝑢,𝑠)=∫_Ω𝐺(𝑢,𝑠)𝑑𝑥, then there exist two positive constants 𝐾₁, 𝐾₂ such that 𝒢𝜑,𝑠+𝑚𝜑²+𝐾₁⩾0,𝜑,𝑔𝜑,𝑠−𝐶₀𝒢𝜑,𝑠+𝑚𝜑²+𝐾₂⩾0,∀𝜑,𝑠∈ℝ×ℝ, (2.7)(2.8)where 𝑚,𝐶₀>0, and 𝑚 is sufficiently small.

By virtue of (2.5), we can get 𝑔(𝑢,𝑠)⩽𝐶₃1+|𝑢|^𝛾+1,|𝐺(𝑢,𝑠)|⩽𝐶₃1+|𝑢|^𝛾+2. (2.9)

When 𝐴=Δ², problem (1.3) is equivalent to the following equations in 𝐻: 𝑢_𝑡𝑡+𝛼𝑢_𝑡+𝐴𝑢+𝑘𝑢⁺()(),𝑢(𝜏)+𝑔𝑢,𝑡=ℎ𝑥,𝑡=𝑢₁,𝑢_𝑡(𝜏)=𝑢₂. (2.10)

From the Poincaré inequality, there exists a proper constant 𝜆₁>0, such that 𝜆₁‖𝑢‖²⩽‖𝑢‖²₁,∀𝑢∈𝑉. (2.11)

We introduce the Hilbert space ℰ₀=𝑉×𝐻,ℰ₁=𝐷(𝐴)×𝑉, (2.12) and endow this space with norm: ‖𝑧‖_ℰ0=𝑢,𝑢_𝑡ℰ0=12‖𝑢‖²₁+𝑢_𝑡^21/2,‖𝑧‖_ℰ1=𝑢,𝑢_𝑡ℰ1=12‖𝑢‖²₂+𝑢_𝑡²₁^1/2. (2.13)

To prove the existence of uniform attractors corresponding to (2.10), we also need the following abstract results (e.g., see [10]).

Let 𝐸 be a Banach space, and let a two-parameter family of mappings {𝑈(𝑡,𝜏)}={𝑈(𝑡,𝜏)∣𝑡⩾𝜏,𝜏∈ℝ} on 𝐸: 𝑈(𝑡,𝜏)∶𝐸→𝐸,𝑡⩾𝜏,𝜏∈ℝ. (2.14)

Definition 2.1 (see [10]). Let Σ be a parameter set. {𝑈_𝜎(𝑡,𝜏)∣𝑡⩾𝜏,𝜏∈ℝ},  𝜎∈Σ is said to be a family of processes in Banach space 𝐸, if for each 𝜎∈Σ,  {𝑈_𝜎(𝑡,𝜏)} is a process; that is, the two-parameter family of mappings {𝑈_𝜎(𝑡,𝜏)} from 𝐸 to 𝐸 satisfy 𝑈_𝜎(𝑡,𝑠)∘𝑈_𝜎(𝑠,𝜏)=𝑈_𝜎()𝑈𝑡,𝜏,∀𝑡⩾𝑠⩾𝜏,𝜏∈ℝ,_𝜎(𝜏,𝜏)=𝐼istheidentityoperator,𝜏∈ℝ, (2.15) where Σ is called the symbol space and 𝜎∈Σ is the symbol.

Note that the following translation identity is valid for a general family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ, if a problem is the unique solvability and for the translation semigroup {𝑇(𝑙)∣𝑙⩾0} satisfying 𝑇(𝑙)Σ=Σ: 𝑈_𝜎(𝑡+𝑙,𝜏+𝑙)=𝑈_{𝑇(𝑙)𝜎}(𝑡,𝜏),∀𝜎∈Σ,𝑡⩾𝜏,𝜏∈ℝ,𝑙⩾0. (2.16)

A set 𝐵₀⊂𝐸 is said to be a uniformly (w.r.t 𝜎∈Σ) absorbing set for the family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ if for any 𝜏∈ℝ and 𝐵∈ℬ(𝐸), there exists 𝑡₀=𝑡₀(𝜏,𝐵)⩾𝜏 such that ∪_𝜎∈Σ𝑈_𝜎(𝑡,𝜏)⊆𝐵₀ for all 𝑡⩾𝑡₀. A set 𝑌⊂𝐸 is said to be uniformly (w.r.t. 𝜎∈Σ) attracting for the family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ if for any fixed 𝜏∈ℝ and every 𝐵∈ℬ(𝐸), lim_𝑡→∞sup_𝜎∈Σdist_𝐸𝑈_𝜎(𝑡,𝜏)𝐵,𝑌=0. (2.17)

Definition 2.2 (see [10]). A closed set 𝐴_Σ⊂𝐸 is said to be the uniform (w.r.t. 𝜎∈Σ) attractor of the family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ if it is uniformly (w.r.t. 𝜎∈Σ) attracting (attracting property) and contained in any closed uniformly (w.r.t. 𝜎∈Σ) attracting set 𝐴^′ of the family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ: 𝐴_Σ⊆𝐴^′ (minimality property).

Now we recalled the results in [16].

Definition 2.3 (see [16, 17]). A family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ is said to be satisfying uniform (w.r.t. 𝜎∈Σ). Condition (C) if for any fixed 𝜏∈ℝ,  𝐵∈ℬ(𝐸) and 𝜖>0, there exist a 𝑡₀=𝑡₀(𝜏,𝐵,𝜖)⩾𝜏 and a finite dimensional subspace 𝐸_𝑚 of 𝐸 such that (i)𝑃_𝑚(∪_𝜎∈Σ∪_𝑡⩾𝑡0𝑈_𝜎(𝑡,𝜏)𝐵) is bounded;(ii)‖(𝐼−𝑃_𝑚)(∪_𝜎∈Σ∪_𝑡⩾𝑡0𝑈_𝜎(𝑡,𝜏)𝑥)‖_𝐸⩽𝜖,∀𝑥∈𝐵, where dim𝐸_𝑚=𝑚 and 𝑃_𝑚∶𝐸→𝐸_𝑚 is abounded projector.

Theorem 2.4 (see [16]). Let Σ be a complete metric space, and let {𝑇(𝑡)} be a continuous invariant 𝑇(𝑡)Σ=Σ semigroup on Σ satisfying the translation identity. A family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ possesses compact uniform (w.r.t. 𝜎∈Σ) attractor 𝐴_Σ in 𝐸 satisfying 𝐴_Σ=𝜔_0,Σ𝐵₀=𝜔_𝜏,Σ𝐵₀,∀𝑡∈ℝ, (2.18) if it (i)has a bounded uniformly (w.r.t. 𝜎∈Σ) absorbing set 𝐵₀;(ii)satisfies uniform (w.r.t. 𝜎∈Σ) condition (C), where 𝜔_𝜏,Σ(𝐵₀)=∩_𝑡⩾𝜏[∪_𝜎∈Σ∪_𝑠⩾𝑡𝑈_𝜎(𝑠,𝑡)𝐵₀]. Moreover, if 𝐸 is a uniformly convex Banach space, then the converse is true.

Let 𝑋 be a Banach space. Consider the space 𝐿²_loc(ℝ;𝑋) of functions 𝜙(𝑠),𝑠∈ℝ with values in 𝑋 that are 2-power integrable in the Bochner sense. 𝐿²_𝑐(ℝ;𝑋) is a set of all translation compact functions in 𝐿²_loc(ℝ;𝑋), 𝐿²_𝑏(ℝ;𝑋) is a set of all translation bound functions in 𝐿²_loc(ℝ;𝑋).

In [15], the authors have introduced a new class of functions which are translation bounded but not translation compact. In the third section, let the forcing term ℎ(𝑥,𝑡) satisfy condition (𝐶^∗), we can prove the existence of compact uniform (w.r.t. 𝜎∈ℋ(𝜎₀), 𝜎₀(𝑠)=(𝑔₀(𝑢,𝑠),ℎ(𝑥,𝑠))) attractor for nonautonomous suspension bridge equation in ℰ₁.

Definition 2.5 (see [15]). Let 𝑋 be a Banach space. A function 𝑓∈𝐿²_𝑏(ℝ;𝑋) is said to satisfy condition (𝐶^∗) if for any 𝜖>0, there exists a finite dimensional subspace 𝑋₁ of 𝑋 such that sup_{𝑡∈ℝ𝑡}^𝑡+1𝐼−𝑃_𝑚𝑓(𝑠)²_𝑋𝑑𝑠<𝜖, (2.19) where 𝑃_𝑚∶𝑋→𝑋₁ is the canonical projector.

Denote by 𝐿²_𝑐∗(ℝ;𝑋) the set of all functions satisfying condition (𝐶^∗). From [15], we can see that 𝐿²_𝑐(ℝ;𝑋)⊂𝐿²_𝑐∗(ℝ;𝑋)⊂𝐿²_𝑏(ℝ;𝑋).

Remark 2.6. In fact, the function satisfying condition (𝐶^∗) implies the dissipative property in some sense, and the condition (𝐶^∗) is very natural in view of the compact condition, uniform condition (𝐶).

Lemma 2.7 (see [15]). If 𝑓∈𝐿²_𝑐∗(ℝ;𝑋), then for any 𝜖>0 and 𝜏∈ℝ we have sup_𝑡⩾𝜏^𝑡_𝜏𝑒^{−𝛿(𝑡−𝑠)}𝐼−𝑃_𝑚𝑓(𝑠)²_𝑋𝑑𝑠⩽𝜖, (2.20) where 𝑃_𝑚∶𝑋→𝑋₁ is the canonical projector and 𝛿 is a positive constant.

In order to define the family of processes of (2.10), we also need the following results.

Proposition 2.8 (see [10]). If 𝑋 is reflexive separable, then (i) for all ℎ₁∈ℋ(ℎ₀), ‖ℎ₁‖_𝐿²_{𝑏(ℝ;𝑋)}⩽‖ℎ₀‖_{𝐿2𝑏(ℝ;𝑋)};(ii) the translation group {𝑇(𝑡)} is weakly continuous on ℋ(ℎ₀);(iii)𝑇(𝑡)ℋ(ℎ₀)=ℋ(ℎ₀) for all 𝑡∈ℝ⁺.

Proposition 2.9 (see [10]). Let 𝑔₀(𝑠)∈𝐿²_𝑐(ℝ;𝑋), then (i) for all 𝑔₁∈ℋ(𝑔₀), 𝑔₁∈𝐿²_𝑐(ℝ;𝑋), and the set ℋ(𝑔₀) is bound in 𝐿²_𝑏(ℝ;𝑋);(ii) the translation group {𝑇(𝑡)} is continuous on ℋ(𝑔₀) with the topology of 𝐿²_loc(ℝ,𝑋);(iii)𝑇(𝑡)ℋ(𝑔₀)=ℋ(𝑔₀) for all 𝑡∈ℝ⁺.

3. Uniform Attractors in ℰ₁

To describe the asymptotic behavior of the solutions of our system, we set ℎ₀∈𝐿²_𝑐∗(ℝ_𝜏;𝑉)⊂𝐿²_𝑏(ℝ_𝜏;𝑉) and ℋ(ℎ₀)=[ℎ₀(𝑥,𝑠+ℎ)∣ℎ∈ℝ]_𝐿^2,𝑤_{loc(ℝ𝜏;𝑉)}, where [] denotes the closure of a set in topological space 𝐿^2,𝑤_loc(ℝ_𝜏;𝑉). If ℎ∈ℋ(ℎ₀), then ℎ∈𝐿²_𝑏(ℝ_𝜏;𝑉), that is to be sup_{𝑡⩾𝜏𝑡}^𝑡+1‖ℎ(𝑥,𝑠)‖₁𝑑𝑠<∞, (3.1) where ‖⋅‖₁ denotes the norm in 𝑉.

3.1. Existence and Uniqueness of Strong Solutions

At first, we give the concept of strong solutions for the initial-boundary value problem (2.10).

Definition 3.1. Set 𝐼=[𝜏,𝑇], for 𝑇>𝜏⩾0. We suppose that 𝑘>0, ℎ∈𝐿²_𝑏(ℝ_𝜏;𝑉), 𝑔∈𝐶¹(ℝ×ℝ;ℝ) satisfying (2.3)–(2.6) and 𝑔(0,0)=0. The function 𝑧=(𝑢,𝑢_𝑡)∈𝐿^∞(𝐼;ℰ₁) is said to be a strong solution to problem (2.10) in the time interval 𝐼, with initial data 𝑧(𝜏)=𝑧_𝜏=(𝑢₁,𝑢₂)∈ℰ₁, provided 𝑢_𝑡𝑡,𝑣+𝛼𝑢_𝑡,𝑣+_ΩΔ𝑢Δ𝑣𝑑𝑥+_Ω𝑔(𝑢,𝑡)𝑣𝑑𝑥+𝑘𝑢⁺,𝑣=_Ωℎ(𝑥,𝑡)𝑣𝑑𝑥, (3.2) for all 𝑣∈𝑉 and a.e. 𝑡∈𝐼.

Then, by using the methods in [18] (Galerkin approximation method), we can get the following result about the existence and uniqueness of strong solutions.

Theorem 3.2 (existence and uniqueness of strong solutions). Define 𝐼=[𝜏,𝑇], for all 𝑇>𝜏. Let 𝑘>0, ℎ∈𝐿²_𝑏(ℝ_𝜏;𝑉), 𝑔∈𝐶¹(ℝ×ℝ;ℝ) satisfying (2.3)–(2.6). Then for any given 𝑧_𝜏∈ℰ₁, there is a unique solution 𝑧=(𝑢,𝑢_𝑡) for problem (2.10) in ℰ₁. Furthermore, for 𝑖=1,2, let {𝑧^𝑖_𝜏,ℎ_𝑖}(𝑧^𝑖_𝜏∈ℰ₁  and  ℎ_𝑖∈𝐿²_𝑏(ℝ_𝜏;𝑉)) be two initial conditions, and denote by 𝑧_𝑖 corresponding solutions to problem (2.10). Then the estimates hold as follows: for all 𝜏⩽𝑡⩽𝑇+𝜏, 𝑧₁(𝑡)−𝑧₂(𝑡)²_ℰ1⩽𝑄𝑧^𝑖_𝜏ℰ1,𝑇𝑧¹_𝜏−𝑧²_𝜏²_ℰ1+ℎ₁−ℎ₂²_{𝐿2𝑏ℝ𝜏;𝑉}. (3.3)

Thus, (2.10) will be written as an evolutionary system introduced 𝑧(𝑡)=(𝑢(𝑡),𝑢_𝑡(𝑡)) and 𝑧_𝜏=𝑧(𝜏)=(𝑢₁,𝑢₂) for brevity, as ‖𝑧‖²_ℰ1=(1/2)(‖𝑢‖²₂+‖𝑢_𝑡‖²₁), the system (2.10) can be written in the operator form 𝜕_𝑡𝑧=𝐴_𝜎(𝑡)(𝑧),𝑧|_𝑡=𝜏=𝑧_𝜏, (3.4) where 𝜎(𝑠)=(𝑔(𝑢,𝑠),ℎ(𝑥,𝑠)) is the symbol of (3.4). If 𝑧_𝜏∈ℰ₁, then problem (3.4) has a unique solution 𝑧(𝑡)∈𝐿^∞(ℝ_𝜏,ℰ₁). This implies that the process {𝑈_𝜎(𝑡,𝜏)} given by the formula 𝑈_𝜎(𝑡,𝜏)𝑧_𝜏=𝑧(𝑡) is defined in ℰ₁.

Now we define the symbol space. A fixed symbol 𝜎₀(𝑠)=(𝑔₀(𝑢,𝑠),ℎ₀(𝑥,𝑠)) can be given, where ℎ₀(𝑥,𝑠) is in 𝐿²_𝑐∗(ℝ_𝜏;𝑉), the function 𝑔₀(𝑢,𝑠)∈𝐿²_𝑐(ℝ_𝜏;ℳ) satisfying (2.3)–(2.6), and ℳ is a Banach space, ℳ=𝑔∈𝐶¹()𝑔(𝑢ℝ×ℝ,ℝ)₁+𝑔_𝑠(𝑢)₁‖𝑢‖₁^𝛾+1+𝑔+1^𝑢(𝑢)₁‖𝑢‖^𝛾₁,+1<∞ (3.5) endowed with the following norm: 𝑔_ℳ=sup_𝑢∈ℝ𝑔(𝑢)₁+𝑔_𝑠(𝑢)₁‖𝑢‖₁^𝛾+1+𝑔+1^𝑢(𝑢)₁‖𝑢‖^𝛾₁.+1 (3.6) Obviously, the function 𝜎₀(𝑠)=(𝑔₀(𝑢,𝑠),ℎ₀(𝑥,𝑠)) is in 𝐿²_𝑐(ℝ_𝜏;ℳ)×𝐿²_𝑐∗(ℝ_𝜏;𝑉). we define ℋ(𝜎₀)=ℋ(𝑔₀)×ℋ(ℎ₀)=[𝑔₀(𝑢,𝑠+𝑙)∣𝑙∈ℝ]_𝐿^2,𝑤_{loc(ℝ𝜏;ℳ)}×[ℎ₀(𝑥,𝑠+𝑙)∣𝑙∈ℝ]_𝐿^2,𝑤_{loc(ℝ𝜏;𝑉)}, where [] denotes the closure of a set in topological space 𝐿^2,𝑤_loc(ℝ_𝜏;ℳ)  (or 𝐿^2,𝑤_loc(ℝ_𝜏;𝑉)). So, if (𝑔,ℎ)∈ℋ(𝜎₀), then 𝑔(𝑢,𝑡) and ℎ(𝑥,𝑡) all satisfy condition (𝐶^∗).

Applying Propositions 2.8 and 2.9 and Theorem 3.2, we can easily know that the family of processes {𝑈_𝜎(𝑡,𝜏)}∶ℰ₁→ℰ₁,  𝜎∈ℋ(𝜎₀), 𝑡⩾𝜏 are defined. Furthermore, the translation semigroup {𝑇(𝑙)∣𝑙∈ℝ⁺} satisfies that for all 𝑙∈ℝ⁺, 𝑇(𝑙)ℋ(𝜎₀)=ℋ(𝜎₀), and the following translation identity: 𝑈_𝜎(𝑡+𝑙,𝜏+𝑙)=𝑈_{𝑇(𝑙)𝜎}(𝑡,𝜏),∀𝜎∈ℋ𝜎₀,for𝑡⩾𝜏⩾0,𝑙⩾0 (3.7) holds.

Then for any 𝜎∈ℋ(𝜎₀), the problem (3.4) with 𝜎 instead of 𝜎₀ possesses a corresponding to process {𝑈_𝜎(𝑡,𝜏)} acting on ℰ₁.

Consequently, for each 𝜎∈ℋ(𝜎₀), 𝜎₀(𝑠)=(𝑔₀(𝑢,𝑠),ℎ₀(𝑥,𝑠)) (here ℎ₀(𝑥,𝑠)∈𝐿²_𝑐∗(ℝ_𝜏;𝑉), 𝑔₀(𝑢,𝑠)∈𝐿²_𝑐(ℝ_𝜏;ℳ) satisfying (2.3)–(2.6)), we can define a process 𝑈_𝜎(𝑡,𝜏)∶ℰ₁→ℰ₁,𝑧_𝜏=𝑢₁,𝑢₂→𝑢(𝑡),𝑢_𝑡(𝑡)=𝑈_𝜎(𝑡,𝜏)𝑧_𝜏, (3.8) and {𝑈_𝜎(𝑡,𝜏)},  𝜎∈ℋ(𝜎₀) is a family of processes on ℰ₁.

3.2. A Priori Estimates

3.2.1. A Priori Estimates in ℰ₀

Theorem 3.3. Assume that 𝑧(𝑡) is a solution of (2.10) with initial data 𝑧₀∈𝐵. If the nonlinearity 𝑔(𝑢,𝑡) satisfies (2.3)–(2.6), ℎ₀∈𝐿²_𝑐∗(ℝ_𝜏;𝐻),  ℎ∈ℋ(ℎ₀), 𝑘>0, then there is a positive constant 𝜇₀ such that for any bounded (in  ℰ₀) subset 𝐵, there exists 𝑡₀=𝑡₀(‖𝐵‖_ℰ0) such that ‖𝑧(𝑡)‖²_ℰ0=12‖𝑢‖²₁+𝑢_𝑡²⩽𝜇²₀,𝑡⩾𝑡₀=𝑡₀‖𝐵‖_ℰ0. (3.9)

Proof. Now we will prove that 𝑧=(𝑢,𝑢_𝑡) are bounded in ℰ₀=𝑉×𝐻.
We assume that 𝜚 is positive and satisfies 0<𝜚𝛼−𝜚<𝜆₁. (3.10) Multiplying (2.10) by 𝑣(𝑡)=𝑢_𝑡(𝑡)+𝜚𝑢(𝑡) and integrating over Ω, we have 12𝑑‖𝑣‖𝑑𝑡²+‖𝑢‖²₁+𝜚‖𝑢‖²₁+𝛼−𝜚‖𝑣‖²−𝜚𝛼−𝜚⟨𝑢,𝑣⟩+𝑘𝑢⁺,𝑣+𝑔(𝑢,𝑡),𝑣=⟨ℎ(𝑡),𝑣⟩. (3.11)
We can easily see that 𝜚⟨⟩⩽‖𝑣‖𝛼−𝜚𝑢,𝑣𝛼−𝜚²4+𝛼−𝜚𝜚²‖𝑢‖²,⟨ℎ(𝑡)⟩⩽‖𝑣‖,𝑣𝛼−𝜚²4+‖ℎ(𝑡)‖².𝛼−𝜚 (3.12)(3.13) Then, substituting (3.12)-(3.13) into (3.11), we can obtain that 𝑑‖𝑣‖𝑑𝑡²+‖𝑢‖²₁+2𝜚‖𝑢‖²₁+𝛼−𝜚‖𝑣‖²−2𝜚²𝛼−𝜚‖𝑢‖²+2𝑘𝑢⁺‖ℎ(𝑡,𝑣+2𝑔,𝑣⩽2)‖².𝛼−𝜚 (3.14) In view of (2.6) and (2.8), we can know 𝑔,𝑣=𝑔,𝑢_𝑡=𝑑+𝜚𝑢𝑑𝑡_Ω𝐺(𝑢(𝑥,𝑡),𝑡)𝑑𝑥+𝜚𝑔(𝑢,𝑡),𝑢−_Ω𝐺_𝑠(𝑢())=𝑑𝑥,𝑡,𝑡𝑑𝑥𝑑𝑡_Ω𝐺(𝑢(𝑥,𝑡),𝑡)𝑑𝑥+𝜚_Ω𝑔(𝑢(𝑥,𝑡),𝑡)𝑢(𝑥,𝑡)𝑑𝑥−𝜚𝐶_0Ω𝐺(𝑢(𝑥,𝑡),𝑡)𝑑𝑥+𝜚𝐶_0Ω𝐺(𝑢(𝑥,𝑡),𝑡)𝑑𝑥−_Ω𝐺_𝑠(𝑢())⩾𝑑𝑥,𝑡,𝑡𝑑𝑥𝒢(𝑢())𝑑𝑡𝑥,𝑡,𝑡+𝜚𝐶₀𝒢(𝑢(𝑥,𝑡),𝑡)−𝜚𝑚‖𝑢‖²+𝐾₂−𝛿²𝒢(𝑢(𝑥,𝑡),𝑡)−𝐶₂|Ω|,𝑘𝑢⁺=1,𝑣2𝑑𝑘𝑢𝑑𝑡⁺²+𝜚𝑘𝑢⁺². (3.15)
Consequently, 𝑑‖𝑣‖𝑑𝑡²+‖𝑢‖²₁+𝑘𝑢⁺²(𝑢())+‖𝑣‖+2𝒢𝑥,𝑡,𝑡𝛼−𝜚²𝜚+2𝜆₁𝜆₁−𝜚𝛼−𝜚−𝑚‖𝑢‖²₁+2𝜚𝑘𝑢⁺²+𝜚𝐶₀−𝛿²(𝑢())‖ℎ(𝑡2𝒢𝑥,𝑡,𝑡⩽2)‖²𝛼−𝜚+2𝜚𝐾₂+𝐶₂|Ω|. (3.16)
We introduce the functional as follows: 𝑦(𝑡)=‖𝑣‖²+‖𝑢‖²₁+𝑘𝑢⁺²+2𝒢(𝑢(𝑥,𝑡),𝑡)+2𝐾₁,for𝑡⩾𝜏. (3.17)
Setting 𝛽=min{𝛼−𝜚,2𝜚𝜆₁⁻¹(𝜆₁−𝜚(𝛼−𝜚)−𝑚),2𝜚,𝜚𝐶₀−𝛿²}, we choose proper positive constants 𝑚 and 𝛿, such that 𝑚<𝜆₁−𝜚𝛼−𝜚,𝛿²<𝜚𝐶₀ (3.18) hold, then 𝛽>0.
We define 𝑚_ℎ(𝑡)=‖ℎ(𝑡)‖², then 𝑑𝑦(𝑡)(𝑡)𝑑𝑡+𝛽𝑦⩽𝐶₄+𝐶₅𝑚_ℎ(𝑡), (3.19) where 𝐶₄=2(𝜚𝐾₂+𝐶₂|Ω|)+2𝛽𝐾₁, 𝐶₅=2(𝛼−𝜚)⁻¹.
Analogous to the proof of Lemma 2.1.3 in [10], we can estimate the integral and obtain 𝑦(𝑡)⩽𝑦(𝜏)𝑒^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹1−𝑒^−𝛽𝑡+𝐶₅^𝑡₀𝑚_ℎ(𝑠)𝑒^{−𝛽(𝑡−𝑠)}(𝜏)𝑒𝑑𝑠⩽𝑦^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹1−𝑒^−𝛽𝑡+𝐶₅^𝑡_𝑡−1𝑚_ℎ(𝑠)𝑒^{−𝛽(𝑡−𝑠)}𝑑𝑠+𝐶₅^𝑡−1_𝑡−2𝑚_ℎ(𝑠)𝑒^{−𝛽(𝑡−𝑠)}(𝜏)𝑒𝑑𝑠+⋯⩽𝑦^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹1−𝑒^−𝛽𝑡+𝐶₅^𝑡_𝑡−1𝑚_ℎ(𝑠)𝑑𝑠+𝐶₅𝑒^{−𝛽𝑡−1}_𝑡−2𝑚_ℎ(𝑠)𝑑𝑠+𝐶₅𝑒^{−2𝛽𝑡−2}_𝑡−3𝑚_ℎ(𝑠)(𝜏)𝑒𝑑𝑠+⋯⩽𝑦^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹1−𝑒^−𝛽𝑡+𝐶₅𝑚_ℎ1+𝑒^−𝛽+𝑒^−2𝛽(𝜏)𝑒+⋯⩽𝑦^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹1−𝑒^−𝛽𝑡+𝐶₅𝑚_ℎ1+𝛽⁻¹⩽𝑦(𝜏)𝑒^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹+𝐶₅𝑚_ℎ1+𝛽⁻¹,for𝑡⩾𝜏, (3.20) where 𝑚_ℎ=sup_𝑡⩾𝜏∫_𝑡^𝑡+1𝑚_ℎ(𝑠)𝑑𝑠.
By virtue of (2.7), we can get 2𝒢(𝑢,𝑡)⩾−2𝑚‖𝑢‖²−2𝐾₁⩾−2𝑚𝜆₁⁻¹‖𝑢‖²₁−2𝐾₁. (3.21) Choosing 𝑚⩽𝜆₁/4, we obtain from (3.17) 𝑦(𝑡)=‖𝑢‖²₁+𝑢_𝑡+𝜚𝑢²+𝑘𝑢⁺²+2𝒢(𝑢,𝑡)+2𝐾₁⩾12‖𝑢‖²₁+𝑢_𝑡+𝜚𝑢²+𝑘𝑢⁺²⩾‖𝑧(𝑡)‖²_ℰ0. (3.22) In consideration of (2.9) and 0<𝛾<∞, we can see 2𝒢𝑢_𝜏(𝑥),𝜏⩽2𝐶_3Ω𝑢_𝜏(𝑥)^𝛾+2+1𝑑𝑥⩽𝐶₆𝑢_𝜏1^𝛾+2,𝑦(𝜏)=‖𝑢(𝜏+1)‖²₁+𝑢_𝑡(𝜏)+𝜚𝑢(𝜏)²+𝑘(𝑢(𝜏))⁺²+2𝒢(𝑢(𝜏),𝜏)+2𝐾₁⩽𝐶₇‖𝑧(𝜏)‖_ℰ^𝛾+2₀+1. (3.23)(3.24)
Combining (3.20), (3.22), and (3.24), we can deduce that ‖𝑧(𝑡)‖²_ℰ0⩽𝑦(𝜏)𝑒^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹+𝐶₅𝑚_ℎ1+𝛽⁻¹⩽𝐶₇‖𝑧(𝜏)‖_ℰ^𝛾+2₀+1𝑒^{−𝛽(𝑡−𝜏)}+𝐶₄𝛽⁻¹+𝐶₅𝑚_ℎ1+𝛽⁻¹⩽𝐶₇‖𝑧(𝜏)‖_ℰ^𝛾+2₀𝑒^{−𝛽(𝑡−𝜏)}+𝐶₈,𝑡⩾𝜏. (3.25)
Assume that ‖𝑧(𝜏)‖²_ℰ0⩽𝑅, as 𝑡⩾𝑡₀=𝑡₀(‖𝐵‖_ℰ0), we have ‖𝑧(𝑡)‖_ℰ0⩽𝜇₀. (3.26)
We complete the proof.

3.2.2. A Priori Estimates in ℰ₁

Lemma 3.4. Assuming that 𝑧(𝑡) is a strong solution of (2.10) with initial data 𝑧₀∈𝐵. If the nonlinearity 𝑔(𝑢,𝑡) satisfies (2.3)–(2.6), ℎ₀∈𝐿²_𝑐∗(ℝ_𝜏;𝑉),  ℎ∈ℋ(ℎ₀), 𝑘>0, then there is a positive constant 𝜇₂ such that for any bounded (in  ℰ₁) subset 𝐵, there exists 𝑡₁=𝑡₁(‖𝐵‖_ℰ1) such that ‖𝑧(𝑡)‖²_ℰ1=12‖𝑢‖²₂+𝑢_𝑡²₁⩽𝜇²₂,𝑡⩾𝑡₁=𝑡₁‖𝐵‖_ℰ1. (3.27)

Proof. Now we will prove that 𝑧=(𝑢,𝑢_𝑡) are bounded in ℰ₁=𝐷(𝐴)×𝑉.
We assume that 𝜚 is positive and satisfies 0<𝜚𝛼−𝜚<𝜆₁. (3.28) Multiplying (2.10) by 𝐴𝑣(𝑡)=𝐴𝑢_𝑡(𝑡)+𝜚𝐴𝑢(𝑡) and integrating over Ω, we have 12𝑑‖𝑣‖𝑑𝑡²¹+‖𝑢‖²₂+𝜚‖𝑢‖²₂+𝛼−𝜚‖𝑣‖²₁−𝜚𝛼−𝜚⟨𝑢,𝑣⟩₁+𝑘𝑢⁺+𝑔()=⟨ℎ(𝑡)⟩,,𝐴𝑣𝑢,𝑡,𝐴𝑣,𝐴𝑣 (3.29) where 𝐴=Δ².
We can deduce that 𝜚𝛼−𝜚⟨𝑢,𝑣⟩₁⩽‖𝑣‖𝛼−𝜚²¹4+𝛼−𝜚𝜚²‖𝑢‖²₁,⟨ℎ(𝑡)⟩⩽‖𝑣‖,𝐴𝑣𝛼−𝜚²¹4+‖ℎ(𝑡)‖²₁.𝛼−𝜚 (3.30) Then, substituting (3.30) into (3.29), we have 𝑑‖𝑣‖𝑑𝑡²¹+‖𝑢‖²₂+2𝜚‖𝑢‖²₂+𝛼−𝜚‖𝑣‖²₁−2𝜚²𝛼−𝜚‖𝑢‖²₁+2𝑘𝑢⁺‖ℎ(𝑡,𝐴𝑣+2𝑔,𝐴𝑣⩽2)‖²₁.𝛼−𝜚 (3.31) In view of (2.5) and Theorem 3.3, we can see that 𝑔,𝐴𝑣=𝑔,𝐴𝑢_𝑡=𝑑+𝜚𝐴𝑢𝑔()−𝑔𝑑𝑡𝑢,𝑡,𝐴𝑢_𝑢(𝑢,𝑡)𝑢_𝑡,𝐴𝑢−𝑔_𝑠()𝑔()⩾𝑑𝑢,𝑡,𝐴𝑢+𝜚𝑢,𝑡,𝐴𝑢𝑔()𝑔()−𝑑𝑡𝑢,𝑡,𝐴𝑢+𝜚𝑢,𝑡,𝐴𝑢_Ω𝑔_𝑢(𝑢,𝑡)⋅𝑢_𝑡⋅||−𝐴𝑢𝑑𝑥_Ω𝑔_𝑠(⋅||⩾𝑑𝑢,𝑡)𝐴𝑢𝑑𝑥𝑔()𝑔()−𝑑𝑡𝑢,𝑡,𝐴𝑢+𝜚𝑢,𝑡,𝐴𝑢_Ω𝐶₁1+𝑢|^𝛾⋅𝑢_𝑡⋅||−𝐴𝑢𝑑𝑥_Ω𝐶₁1+|𝑢|^𝑦+1⋅||⩾𝑑𝐴𝑢𝑑𝑥𝑔()𝑔()‖𝑢‖𝑑𝑡𝑢,𝑡,𝐴𝑢+𝜚𝑢,𝑡,𝐴𝑢−𝐶₂⩾𝑑𝑔()𝑔()−𝜚𝑑𝑡𝑢,𝑡,𝐴𝑢+𝜚𝑢,𝑡,𝐴𝑢8‖𝑢‖²₂−𝐶. (3.32) Exploiting ‖(𝑢⁺)_𝑡‖⩽‖𝑢_𝑡‖ and Theorem 3.3, we can obtain 𝑘𝑢⁺,𝐴𝑣=𝑘𝑢⁺,𝐴𝑢_𝑡=𝑑+𝜚𝐴𝑢𝑘𝑢𝑑𝑡⁺,𝐴𝑢−𝑘𝑢⁺_𝑡,𝐴𝑢+𝜚𝑘𝑢⁺⩾𝑑,𝐴𝑢𝑘𝑢𝑑𝑡⁺,𝐴𝑢−𝑘𝑢⁺_𝑡⋅‖𝐴𝑢‖+𝜚𝑘𝑢⁺⩾𝑑,𝐴𝑢𝑘𝑢𝑑𝑡⁺,𝐴𝑢−𝑘𝑢_𝑡⋅‖𝐴𝑢‖+𝜚𝑘𝑢⁺⩾𝑑,𝐴𝑢𝑘𝑢𝑑𝑡⁺,𝐴𝑢−𝑘𝜇₀‖𝐴𝑢‖+𝜚𝑘𝑢⁺⩾𝑑,𝐴𝑢𝑘𝑢𝑑𝑡⁺−𝜚,𝐴𝑢8‖𝑢‖²₂−𝐶+𝜚𝑘𝑢⁺,𝐴𝑢. (3.33) Consequently, 𝑑‖𝑣‖𝑑𝑡²¹+‖𝑢‖²₂+2𝑘𝑢⁺,𝐴𝑢+2𝑔(𝑢,𝑡),𝐴𝑢+𝛼−𝜚‖𝑣‖²₁𝜚+2𝜆₁34𝜆₁−𝜚𝛼−𝜚‖𝑢‖²₂+2𝜚𝑘𝑢⁺𝑔()‖ℎ(𝑡,𝐴𝑢+2𝜚𝑢,𝑡,𝐴𝑢⩽2)‖²₁𝛼−𝜚+𝐶. (3.34) Choose 𝜚 small enough such that 32−2𝜚𝛼−𝜚𝜆₁⩾1. (3.35) This leads to 𝑑‖𝑣‖𝑑𝑡²¹+𝐴𝑢+𝑘𝑢⁺+𝑔(𝑢,𝑡)²+𝛼−𝜚‖𝑣‖²₁+𝜚𝐴𝑢+𝑘𝑢⁺+𝑔(𝑢,𝑡)²‖ℎ(𝑡⩽2)‖²₁𝛼−𝜚+𝐶+2_Ω𝑔(𝑢,𝑡)𝑔_𝑢(𝑢,𝑡)𝑢_𝑡+𝑔_𝑠(𝑢,𝑡)𝑑𝑥+2𝑘_Ω𝑢⁺⋅𝑢⁺_𝑡𝑔𝑑𝑥+2𝑘_𝑢(𝑢,𝑡)𝑢_𝑡+𝑔_𝑠(𝑢,𝑡),𝑢⁺+2𝑘𝑔(𝑢,𝑡),𝑢⁺𝑢⁺_𝑡+𝜚𝑔(𝑢)²+𝜚𝑘²𝑢⁺²+2𝜚𝑘𝑔(𝑢,𝑡),𝑢⁺. (3.36) By (2.5), (2.9), the Hölder inequality, and Theorem 3.3, we have 𝑑‖𝑣‖𝑑𝑡²¹+𝐴𝑢+𝑘𝑢⁺+𝑔(𝑢,𝑡)²+𝛼−𝜚‖𝑣‖²₁+𝜚𝐴𝑢+𝑘𝑢⁺+𝑔(𝑢,𝑡)²‖ℎ(𝑡⩽2)‖²₁𝛼−𝜚+𝐶. (3.37)
We introduce the functional as follows: 𝒱(𝑡)=‖𝑣‖²₁+𝐴𝑢+𝑘𝑢⁺+𝑔(𝑢,𝑡)²,for𝑡⩾𝜏. (3.38)
Setting 𝛽=min{𝛼−𝜚,𝜚}, we define 𝑚^∗_ℎ(𝑡)=‖ℎ(𝑡)‖²₁, then 𝑑𝒱(𝑡)(𝑡)𝑑𝑡+𝛽𝒱⩽𝐶+𝐶₅𝑚^∗_ℎ(𝑡), (3.39) where 𝐶₅=2(𝛼−𝜚)⁻¹.
Analogous to the proof of Theorem 3.3, we can estimate the integral and obtain 𝒱(𝑡)⩽𝒱(𝜏)𝑒^{−𝛽(𝑡−𝜏)}+𝐶𝛽⁻¹1−𝑒^−𝛽𝑡+𝐶₅^𝑡₀𝑚^∗_ℎ(𝑠)𝑒^{−𝛽(𝑡−𝑠)}(𝜏)𝑒𝑑𝑠⩽𝒱^{−𝛽(𝑡−𝜏)}+𝐶𝛽⁻¹+𝐶₅𝑚^∗_ℎ1+𝛽⁻¹,for𝑡⩾𝜏, (3.40) where 𝑚^∗_ℎ=sup_𝑡⩾𝜏∫_𝑡^𝑡+1𝑚^∗_ℎ(𝑠)𝑑𝑠.
Assuming that ‖𝒱(𝜏)‖_ℰ1⩽𝑅, as 𝑡⩾𝑡₁=𝑡₁(‖𝐵‖_ℰ1), we have ‖𝒱(𝑡)‖_ℰ1⩽𝜇²₁. (3.41) Applying (2.9), the Hölder inequality, the Cauchy inequality, and Theorem 3.3, we can deduce from (3.41) that ‖𝐴𝑢‖²+𝑢_𝑡²₁⩽𝜇²₂, (3.42) where 𝜇₂ depends on 𝜚, 𝛼, 𝑘, ‖ℎ‖²₁, 𝜇₀, and 𝜇₁.
We complete the proof.

And then, combining Theorem 3.2 with Lemma 3.4, we can get the result as follows.

Theorem 3.5 (bounded uniformly absorbing set in ℰ₁). Presuming that 𝑔₀∈𝐿²_𝑐(ℝ_𝜏;𝑀) and ℎ₀∈𝐿²_𝑐∗(ℝ_𝜏;𝑉). Let 𝑔∈ℋ(𝑔₀) satisfy (2.3)–(2.6), ℎ∈ℋ(ℎ₀), and {𝑈_𝜎(𝑡,𝜏)},  𝜎∈ℋ(𝜎₀)=ℋ(𝑔₀)×ℋ(ℎ₀) be the family of processes corresponding to (2.10) in ℰ₁, then {𝑈_𝜎(𝑡,𝜏)} has a uniformly (w.r.t. 𝜎∈ℋ(𝜎₀)) absorbing set 𝐵₁=𝐵_ℰ1(0,𝜇₂) in ℰ₁. That is, for any bounded subset 𝐵⊂ℰ₁, there exists 𝑡₁=𝑡₁(‖𝐵‖_ℰ1) such that _𝜎𝜎∈∪_ℋ0𝑈_𝜎(𝑡,𝜏)𝐵⊂𝐵₁,∀𝑡⩾𝑡₁. (3.43)

3.3. The Existence of Uniform Attractor

We will show the existence of uniform attractor to problem (2.10) in ℰ₁.

Theorem 3.6 (uniform attractor). Let {𝑈_𝜎(𝑡,𝜏)} be the family of processes corresponding to problem (2.10). If 𝑔₀∈𝐿²_𝑐(ℝ_𝜏;ℳ) satisfyies (2.3)–(2.6), ℎ₀∈𝐿²_𝑐∗(ℝ_𝜏;𝑉), and 𝜎₀=(𝑔₀,ℎ₀), then {𝑈_𝜎(𝑡,𝜏)} possesses a compact uniform (w.r.t. 𝜎∈ℋ(𝜎₀)) attractor 𝒜_ℋ(𝜎0) in ℰ₁, which attracts any bounded set in ℰ₁ with ‖⋅‖_ℰ1, satisfying 𝒜_ℋ(𝜎0)=𝜔_0,ℋ(𝜎0)𝐵₁=𝜔_{𝜏,ℋ(𝜎0)}𝐵₁, (3.44) where 𝐵₁ is the uniformly (w.r.t. 𝜎∈ℋ(𝜎₀)) absorbing set in ℰ₁.

Proof. From Theorems 2.4 and 3.5, we merely need to prove that the family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈ℋ(𝜎₀) satisfies uniform (w.r.t. 𝜎∈ℋ(𝜎₀)) condition (𝐶) in ℰ₁. We assume that 𝜆_𝑖,  𝑖=1,2,… are eigenvalue of operator 𝐴 in 𝐷(𝐴), satisfying 0<𝜆₁<𝜆₂⩽⋯⩽𝜆_𝑗⩽⋯,𝜆_𝑗→∞,as𝑗→∞, (3.45)𝜔_𝑖 denotes eigenvector corresponding to eigenvalue 𝜆_𝑖,  𝑖=1,2,3,…, which forms an orthogonal basis in 𝐷(𝐴); at the same time they are also a group of canonical basis in 𝑉 or 𝐻, and satisfy 𝐴𝜔_𝑖=𝜆_𝑖𝜔_𝑖,∀𝑖∈ℕ. (3.46)
Let 𝑉_𝑚=span{𝜔₁,𝜔₂,…,𝜔_𝑚},  𝑃_𝑚∶𝑉→𝑉_𝑚 is an orthogonal projector. For any (𝑢,𝑢_𝑡)∈ℰ₁, we write 𝑢,𝑢_𝑡=𝑢₁,𝑢_1𝑡+𝑢₂,𝑢_2𝑡, (3.47) where (𝑢₁,𝑢_1𝑡)=(𝑃_𝑚𝑢,𝑃_𝑚𝑢_𝑡).
Choose 0<𝜚<1, and 0<𝜚(𝛼−𝜚)<(1/2)𝜆₁. Taking the scalar product with 𝐴𝑣₂(𝑡)=𝐴𝑢_2𝑡(𝑡)+𝜚𝐴𝑢₂(𝑡) for (2.10) in 𝐻, we have 12𝑑𝑣𝑑𝑡²²¹+𝑢₂²₂+𝜚𝑢₂²₂−𝜚𝛼−𝜚𝑢,𝑣₂₁+𝛼−𝜚𝑣₂²₁+𝑘𝑢⁺,𝐴𝑣₂+𝑔(𝑢,𝑡),𝐴𝑣₂=ℎ(𝑡),𝐴𝑣₂, (3.48) where ℎ(𝑡),𝐴𝑣₂⩽𝛼−𝜚𝑣₂²₁8+2𝛼−𝜚⁻¹𝐼−𝑃_𝑚ℎ(𝑡)²₁,−𝑔(𝑢,𝑡),𝐴𝑣₂⩽𝛼−𝜚𝑣₂²₁8+2𝛼−𝜚⁻¹𝐼−𝑃_𝑚𝑔(𝑢,𝑡)²₁. (3.49)(3.50)
Clearly, we can get that 𝜚𝛼−𝜚𝑢,𝑣₂₁⩽𝛼−𝜚𝑣₂²₁4+𝛼−𝜚𝜚²𝑢₂²₁,𝑘𝑢⁺,𝐴𝑣₂=𝑘𝑢⁺,𝐴𝑢_𝑡=𝑑+𝜚𝐴𝑢𝑘𝑢𝑑𝑡⁺,𝐴𝑢−𝑘𝑢⁺_𝑡,𝐴𝑢+𝜚𝑘𝑢⁺⩾1,𝐴𝑢2𝑑𝑘𝑢𝑑𝑡²⁺²¹+𝜚𝑘𝑢₂⁺²₁−𝜚2‖𝑢‖²₂−𝐶. (3.51)(3.52)
Combining (3.49)–(3.52), we obtain from (3.48) 12𝑑𝑣𝑑𝑡²²¹+𝑢₂²₂+𝑘𝑢₂⁺²₁+12𝜚𝑢₂²₂+12𝛼−𝜚𝑣₂²₁+𝜚𝑘𝑢₂⁺²₁−𝛼−𝜚𝜚²𝑢₂²₁⩽12𝑑𝑣𝑑𝑡²²¹+𝑢₂²₂+𝑘𝑢₂⁺²₁+𝜚𝜆₁⁻¹12𝜆₁−𝛼−𝜚𝜚𝑢₂²₂+12𝛼−𝜚𝑣₂²₁+𝜚𝑘𝑢₂⁺²₁⩽2𝛼−𝜚⁻¹𝐼−𝑃_𝑚𝑔(𝑢,𝑡)²₁+2𝛼−𝜚⁻¹𝐼−𝑃_𝑚ℎ(𝑡)²₁⩽2𝐶𝛼−𝜚⁻¹𝐼−𝑃_𝑚𝑔(𝑢,𝑡)²_ℳ1+𝑢₂₁^2𝛾+2+2𝛼−𝜚⁻¹𝐼−𝑃_𝑚ℎ(𝑡)²₁. (3.53)
We define the functional ℒ(𝑡)=12𝑣₂²₁+𝑢₂²₂+𝑘𝑢₂⁺²₁, (3.54) and set 𝜔=min{2𝜚𝜆₁⁻¹((1/2)𝜆₁−(𝛼−𝜚)𝜚),𝛼−𝜚,2𝜚}, then 𝑑ℒ(𝑡)(𝑡)𝑑𝑡+𝜔ℒ⩽2𝐶𝛼−𝜚⁻¹𝐼−𝑃_𝑚𝑔(𝑢,𝑡)²_ℳ1+√2𝜇₁^2𝛾+2+2𝛼−𝜚⁻¹𝐼−𝑃_𝑚ℎ(𝑡)²₁,for𝑡⩾𝑡₁. (3.55) By Gronwall Lemma, we obtain ℒ(𝑡)⩽ℒ𝑡₁𝑒^{−𝜔(𝑡−𝑡1)}+2𝛼−𝜚^𝑡_𝑡1𝑒^{−𝜔(𝑡−𝑠)}𝐼−𝑃_𝑚ℎ(𝑠)²₁+𝑑𝑠2𝐶𝛼−𝜚^𝑡_𝑡1𝑒^{−𝜔(𝑡−𝑠)}𝐼−𝑃_𝑚𝑔(𝑢,𝑠)²_ℳ𝑑𝑠,for𝑡⩾𝑡₁. (3.56)
Obviously, there exists a constant 𝐶, such that 𝑧₂(𝑡)²_ℰ1⩽ℒ(𝑡)⩽𝐶𝑧₂(𝑡)²_ℰ1, (3.57) so 𝑧₂(𝑡)²_ℰ1⩽𝐶𝑧₂𝑡₁²_ℰ1𝑒^{−𝜔(𝑡−𝑡1)}+2𝛼−𝜚^𝑡_𝑡1𝑒^{−𝜔(𝑡−𝑠)}𝐼−𝑃_𝑚ℎ(𝑠)²₁+𝑑𝑠2𝐶𝛼−𝜚^𝑡_𝑡1𝑒^{−𝜔(𝑡−𝑠)}𝐼−𝑃_𝑚𝑔(𝑢,𝑠)²_ℳ𝑑𝑠. (3.58)
Since 𝑔∈𝐿²_𝑐(ℝ_𝜏,ℳ)⊂𝐿²_𝑐∗(ℝ_𝜏,ℳ), ℎ∈𝐿²_𝑐∗(ℝ_𝜏,𝐻), from Lemma 2.7, we can know for any 𝜖₁>0, there exists a constant 𝑚 large enough such that 2𝛼−𝜚^𝑡_𝑡1𝑒^{−𝜔(𝑡−𝑠)}𝐼−𝑃_𝑚ℎ(𝑠)²₁𝜖𝑑𝑠⩽¹3,∀ℎ∈ℋℎ₀,2𝐶𝛼−𝜚^𝑡_𝑡1𝑒^{−𝜔(𝑡−𝑠)}𝐼−𝑃_𝑚𝑔(𝑢,𝑠)²_ℳ𝜖𝑑𝑠⩽¹3,∀𝑔∈ℋ𝑔₀, (3.59) where 𝑡⩾𝜏.
Let 𝑡₂=1/𝜔ln(3𝐶𝜇²₁/𝜖₁)+𝑡₁, then 𝐶𝑧₂𝑡₁²_ℰ1𝑒^{−𝜔(𝑡−𝑡1)}⩽𝜖₁3,∀𝑡⩾𝑡₂. (3.60)
So for every 𝜎∈ℋ(𝜎₀), we can get 𝑧₂(𝑡)²_ℰ1⩽𝜖₁,∀𝑡⩾𝑡₂, (3.61) where ‖𝑧₂(𝑡)‖²_ℰ1=(1/2)(‖𝑢₂‖²₂+‖𝑢_2𝑡‖²₁).
Therefore, the family of processes 𝑈_𝜎(𝑡,𝜏),  𝜎∈ℋ(𝜎₀) satisfies uniformly (w.r.t. 𝜎∈ℋ(𝜎₀)) condition (𝐶) in ℰ₁. Applying Theorem 2.4, we can obtain the existence of uniform (w.r.t. 𝜎∈ℋ(𝜎₀)) attractor of the family of processes 𝑈_𝜎(𝑡,𝜏),  𝜎∈ℋ(𝜎₀) in ℰ₁, which satisfies (3.44).
We complete the proof.

Acknowledgment

This work was partly supported by the NSFC 11101334, the NSFC 11261053, and the NWNU-LKQN-11-5.