Strong Attractor of Beam Equation with Structural Damping and Nonlinear Damping

Abstract

This paper is mainly concerned with the existence of a global strong attractor for the nonlinear extensible beam equation with structural damping and nonlinear external damping. This kind of problem arises from the model of an extensible vibration beam. By the asymptotic compactness of the related continuous semigroup, we prove the existence of a strong global attractor which is connected with phase space 𝐷(Δ²)×𝐻¹₀(Ω)∩𝐻²(Ω).

1. Introduction

Global attractor is a basic concept in the study of long-time behavior of nonlinear dissipative evolution equations with various dissipation. There have been many methods to prove the existence of the global attractor. It can be proved by the theory of 𝛼-contractions of the solution semigroup 𝑆(𝑡), such as [1–3] and the reference therein. It can also be proved by the decomposition of the solution semigroup 𝑆(𝑡) (see Hale [4], Temam [5], etc.).

In this paper, we use the method of the asymptotically compact property of the solution semigroup 𝑆(𝑡) which is different from the method of [1–5] to prove the existence of a strong global attractor for the Kirchhoff type equations with structural damping and nonlinear external damping which arises from the model of the nonlinear vibration beam 𝑢_𝑡𝑡+𝛼Δ²𝑢+𝛾Δ²𝑢_𝑡−𝛽+𝑀_Ω|∇𝑢|²𝑑𝑥+𝑁_Ω∇𝑢∇𝑢_𝑡(𝑢)𝑢𝑑𝑥Δ𝑢+𝑔+𝑓_𝑡=ℎ(𝑥),inΩ×𝑅⁺,𝑢=Δ𝑢=0on𝜕Ω×𝑅⁺,𝑢(𝑥,0)=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥)inΩ, (1)(2)(3) where 𝛼,𝛾, and 𝛽 are all positive constants, Ω is a bounded domain of 𝑅^𝑁 with smooth boundary Γ=𝜕Ω, 𝑀(𝑠), 𝑁(𝑠), 𝑔(𝑠), and 𝑓(𝑠) are nonlinear functions specified later, and ℎ∈𝐿²(Ω) is an external force term. 𝑢(𝑡) represents the vertical deflection of the beam, and 𝑢=𝑢(𝑥,𝑡) is a real-valued function on Ω×[0,+∞).

In this context of problem (1), based on the vibrating beams equation 𝑢_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑥}−𝛼+𝛽^𝑙₀𝑢_𝑥(𝑠,𝑡)²𝑑𝑠𝑢_𝑥𝑥=0 (4) which is proposed by Woinowsky-krieger [6]; Ma and Narciso [7] considered problem (1) without structural damping and posed a weak global attractor in weak phase space 𝐻²₀(Ω)×𝐿²(Ω). Eden and Milani [8] considered the existence of exponential attractor for problem (1) with 𝑓(𝑢)=0 and a linear weak damping 𝑔(𝑢_𝑡)=𝑢_𝑡, 𝑀(⋅) being a nonlinear function and without structural damping. Ball [9] presented the existence and uniqueness of global solutions for problem (1) with 𝑓=𝑔=ℎ=0,𝑀(⋅),𝑁(⋅) are all linear functions.

On the other hand, the existence of the attractor for a related problem, with the boundary conditions 𝑢=Δ𝑢=0 of (2) replaced with 𝑢=∇𝑢=0, was considered by Ma and Narciso [7], Eden and Milani [8] with a linear damping 𝑢_𝑡 or nonlinear damping 𝑓(𝑢_𝑡) without structural damping, respectively. Chueshov and Lasiecka [10] considered a kind of boundary condition which is 𝑢=Δ𝑢=0 but without structural damping.

Generally speaking, there have been many works on the long-time behavior for nonlinear beam equations [6–10]. But for the beam equation (1) with structural damping, in strong phase space 𝐷(Δ²)×𝐻¹₀(Ω)∩𝐻²(Ω), the global solutions and the strong global attractor have not still been proved until now.

The outline of this paper is arranged as follows: in Section 2 we give the existence and uniqueness of global solutions in space 𝐶(𝑅⁺;𝐷(Δ²)×𝐻¹₀(Ω)∩𝐻²(Ω)), in Section 3 we give the boundedness of solutions in phase space 𝐷(Δ²)×𝐻¹₀(Ω)∩𝐻²(Ω), and finally in Section 4, we give the proof of the existence of a strong global attractor in phase space 𝐷(Δ²)×𝐻¹₀(Ω)∩𝐻²(Ω).

2. Some Assumptions and Existence of Global Solution

In (1), we assume that damping term and the source term are in the form of 𝑓𝑢_𝑡=𝑢_𝑡^𝑟𝑢_𝑡,𝑔(𝑢)=|𝑢|^𝜌𝑢 (5) with 20<𝜌,𝑟≤𝑁−2if𝑁≥3,𝜌,𝑟>0if𝑁=1,2. (6)

We assume that the nonlinear functions 𝑀,𝑁∶𝑅⁺→𝑅⁺ are all class 𝐶¹, and satisfying 𝑀(0)=0,𝑁(0)=0 and 𝑀(𝑠)𝑠≥𝑀(𝑠),where𝑀(𝑠)=^𝑠₀𝑀(𝑧)𝑀(𝑠)𝑁(𝑠)𝑑𝑧,≥2𝑠;≥𝑠,∀𝑠∈𝑅. (7)

The functions 𝑓,𝑔∶𝑅→𝑅 are also class 𝐶¹, with 𝑓(0)=𝑔(0)=0, 𝛼₁≤𝑓^′(𝑣)≤𝛼₂, and |𝑔^′(𝑢)|≤𝑘₀(1+|𝑢|^𝜌) for all 𝑢,𝑣∈𝑅, where 𝛼₁,𝛼₂, and 𝑘₀ are all constants. There also exists constants 𝑘₅,𝑘₆ such that 𝑓(𝑢)−𝑓(𝑣)≤𝑘₅1+|𝑢|^𝑟+|𝑣|^𝑟||𝑔(𝑢)(𝑣𝑢−𝑣,∀𝑢,𝑣∈𝑅,−𝑔)≤𝑘₆1+|𝑢|^𝜌+|𝑣|^𝜌|𝑢−𝑣|,∀𝑢,𝑣∈𝑅. (8)

In addition, nonlinear function 𝑔(⋅) also satisfies 𝜑(𝑢)+𝛼−𝜀𝛾8‖𝑢‖²≥−𝑘₁,_Ω𝑔(𝑢)𝑢𝑑𝑥−𝐶₁𝜑(𝑢)+𝛼4‖𝑢‖²≥−𝑘₂, (9) where 𝜑(𝑢)=∫_Ω𝐺(𝑢)𝑑𝑥, 𝐺(𝑢)=∫_Ω𝑔(𝑢)𝑑𝑢, and 𝑘₁,𝑘₂ are all constants, 𝐶₁≥1.

Our analysis is based on the following Sobolev spaces: 𝐻=𝐿²(Ω), 𝑉=𝐻¹₀(Ω)∩𝐻²(Ω), with the usual inner products and norms as follows, respectively: (𝑢,𝑣)=_Ω𝑢𝑣𝑑𝑥,|𝑢|=(𝑢,𝑢)^1/2,∀𝑢,𝑣∈𝐿²(Ω),(Δ𝑢,Δ𝑣)=_ΩΔ𝑢Δ𝑣𝑑𝑥,‖𝑢‖=(Δ𝑢,Δ𝑢)^1/2,∀𝑢,𝑣∈𝐻¹₀(Ω)∩𝐻²(Ω). (10)

Consider 𝐷(Δ²)={𝑢∣𝑢∈𝑉,𝑢∈𝐻⁴(Ω),Δ²𝑢∈𝐻,Δ𝑢|_𝜕Ω=0} with the inner products (Δ²𝑢,Δ²𝑢) and the norms |Δ²𝑢|²=(Δ²𝑢,Δ²𝑢).

Take 𝐸₀=𝐻¹₀(Ω)∩𝐻²(Ω)×𝐿²(Ω) and 𝐸=𝐷(Δ²)×𝐻²(Ω)∩𝐻¹₀(Ω) with the inner products and norms as follows, respectively: 𝑦₁,𝑦_2𝐸0=Δ𝑢₁,Δ𝑢₂+𝑣₁,𝑣₂,𝑦_𝐸0=𝑦,𝑦_𝐸^1/2₀,∀𝑦_𝑖=𝑢_𝑖,𝑣_𝑖^𝑇,𝑦=(𝑢,𝑣)^𝑇∈𝐸₀𝑦,𝑖=1,2,₁,𝑦_2𝐸=Δ²𝑢₁,Δ²𝑢₂+Δ𝑣₁,Δ𝑣₂,𝑦_𝐸=𝑦,𝑦_𝐸^1/2,∀𝑦_𝑖=𝑢_𝑖,𝑣_𝑖^𝑇,𝑦=(𝑢,𝑣)^𝑇∈𝐸,𝑖=1,2. (11)

Note that assumption (6) implies that 𝐻¹₀(Ω)∩𝐻²(Ω)↪𝐻¹₀(Ω)↪𝐿^2(𝑝+1)(Ω), with 𝑝=𝜌 or 𝑝=𝑟.

Finally, we assume that 𝜆,𝜎 are the first eigenvalue of Δ² and Δ, respectively; then we have ‖𝑢‖²≥𝜎|𝑢|²Δ,∀𝑢∈𝑉,₂𝑢²≥𝜆‖𝑣‖²,∀𝑢∈𝐷Δ². (12) In the following, we state the result of the existence and uniqueness of the solutions for systems (1)–(3).

Theorem 1. Assume that (𝑢₀,𝑢₁)∈𝐸, ℎ∈𝐿²(Ω), and the assumptions of these functions 𝑀(⋅),𝑁(⋅),𝑓(⋅), and 𝑔(⋅) hold; then problems (1)–(3) have unique solutions (𝑢,𝑢_𝑡)∈𝐶([0,𝑇];𝐷(Δ²))×𝐶([0,𝑇];𝐻¹₀(Ω)∩𝐻²(Ω)) depending continuously on initial data in 𝐸.

By virtue of Galerkin method, we may prove Theorem 1 combined with the priori estimates of Section 3.

According by Theorem 1, for any 𝑡>0, we may introduce the mapping {𝑆(𝑡),𝑡≥0}∶𝑢₀,𝑢₁⟶𝑢(𝑡),𝑢_𝑡(𝑡). (13) It maps 𝐸 into itself, and it enjoys the usual semigroup properties as follows: 𝑆(0)𝑆()(𝑡)𝑆(𝜏)=𝐼,𝑡+𝜏=𝑆,∀𝑡≥0. (14) And it is obvious that the map {𝑆(𝑡),𝑡>0}, for all 𝑡∈𝑅, is continuous in space 𝐸. In the following, we will introduce the existence of bounded absorbing set and global attractor in space 𝐸 for map {𝑆(𝑡),𝑡≥0}.

3. The Existence of Bounded Absorbing Set in Space 𝐸

In this section, we will show boundedness of the solutions for systems (1)–(3).

Theorem 2. Assume that these assumptions of Theorem 1 hold then for the dynamic system determined by problems (1)–(3), there exists the boundary absorbing set in space 𝐸.

Proof. Taking the inner products of 𝑣=𝑢_𝑡+𝜀𝑢 with both sides of (1) and then making summation, we have 12𝑑𝐸(𝑡)|𝑣|𝑑𝑡−𝜀²+𝜀𝛼−𝜀𝛾‖𝑢‖²+𝜀𝛽|∇𝑢|²+𝛾|𝑣|²+𝜀²()𝑀(𝑧)(𝑧)̇̇𝑔(𝑢)+𝑓𝑢𝑢,𝑣+𝜀+𝑁𝑧+,𝑣_𝑡,𝑣=(ℎ,𝑣), (15) where 𝑀(𝑧)=∫^𝑧₀𝑀(𝑧)𝑑𝑧, 𝑧(𝑡)=|∇𝑢|² and 𝜀 is fixed at arbitrary time, and here the energy function 𝐸(𝑡) is defined on 𝐸₀ by 𝐸(𝑡)=|𝑣|²+𝛼−𝜀𝛾‖𝑢‖²+𝛽|∇𝑢|²+𝑀(𝑧)+|∇𝑢|⁴. (16) Considering the assumption ∫_Ω𝑢𝑔(𝑢)𝑑𝑥−𝐶₁𝜑(𝑢)+(𝛼/4)‖𝑢‖²≥−𝑘₂, we have 𝑔(𝑢)=𝑑,𝑣𝜑(𝑢)𝑔(𝑢)≥𝑑𝑑𝑡+𝜀,𝑢𝜑(𝑢)𝑑𝑡+𝜀𝐶₁𝜑(𝑢)−𝜀𝛼4‖𝑢‖²−𝜀𝑘₂. (17) With |𝜀²(𝑢,𝑣)|≤(𝜀²/𝜎²)‖𝑢‖²+(𝜀²/4)|𝑣|², we have −𝜀|𝑣|²+𝜀𝛼−𝜀𝛾‖𝑢‖²+𝜀𝛽|∇𝑢|²+𝛾|𝑣|²+𝜀²()≥𝑢,𝑣𝜀𝛼−𝜀²𝜀𝛾−²𝜎²‖𝑢‖²+𝛾𝜆²𝜀−𝜀−²4|𝑣|²+𝜀𝛽|∇𝑢|². (18) With the assumptions 𝑓(0)=0,𝑓∈𝐶¹(𝑅,𝑅), and 𝛼₁≤𝑓^′(𝑣)≤𝛼₂ and by using Mean Value Theorem and Mean Value inequality, we have 𝑓𝑢_𝑡,𝑣=𝑓^′(𝜉)𝑣²𝑑𝑥−𝜀𝑓^′(𝜉)≥𝛼𝑢𝑣𝑑𝑥₁−𝜀𝛼₂2|𝑣|²−𝜀𝛼₂2𝜎²‖𝑢‖², (19) where 𝜉 among 0 and 𝑣−𝜀𝑢. Set 𝐸(𝑡)=|𝑣|²+𝛼−𝜀𝛾‖𝑢‖²+𝛽|∇𝑢|²+𝑀(𝑧)+|∇𝑢|⁴+2𝜑(𝑢)+2𝑘₁. (20) Consider 𝑌(𝑡)=𝜀𝛼−𝜀²𝜀𝛾−²𝜎²−𝜀𝛼4−𝜀𝛼₂2𝜎²‖𝑢‖²+𝛾𝜆²𝜀−𝜀−²4+𝛼₁−𝛾𝜎²4−𝜀𝛼₂2|𝑣|²+𝜀𝛽|∇𝑢|²+𝜀𝑀(𝑧)+𝑁(𝑧)̇𝑧̇+𝜀𝐶₁𝜑(𝑢)+𝜀𝑘₁. (21) So (15) is transformed into 12𝑑𝐸(𝑡)(𝑡)≤1𝑑𝑡+𝑌𝛾𝜎²|ℎ|²+𝜀𝑘₂+𝜀𝑘₁. (22) Considering the assumptions 𝑀(𝑠)𝑠≥𝑀(𝑠), 𝑀(𝑠)≥2𝑠, 𝑁(𝑠)>𝑠, ‖𝑢‖²≥𝜎²|𝑢|², and |Δ²𝑢|²≥𝜆²‖𝑢‖², 𝐶₁≥1 and letting 0<𝜀<min{(𝛼𝜎²+2𝛼₂)/(2𝛾𝜎²+4),−(3+𝛼₂)+(3+𝛼₂)²+(4𝛼₁+3𝛾𝜎²)}=𝜀₀, we have 2𝜀𝑌(𝑡)−𝐸(𝑡)>0. (23) Substituting (23) into (22), we have 12𝑑𝐸(𝑡)+𝜀𝑑𝑡2𝐸(𝑡)≤1𝛾𝜎²|ℎ|²+𝜀𝑘₂+𝜀𝑘₁. (24) On the one hand, applying the Gronwall inequality to (24), we get 𝐸(𝑡)≤𝐸(0)𝑒^−𝜀𝑡+2𝜀1𝛾𝜎²|ℎ|²+𝜀𝑘₂+𝜀𝑘₁,𝑡≥0. (25) Note that ‖𝑢(0)‖ and |𝑢_𝑡(0)| are bounded; then there exists a positive constant 𝑅>0 such that 𝐸(0)≤𝑅² is bounded; so limsup_𝑡→∞𝐸(𝑡)≤𝜌²₀=2𝜀1𝛾𝜎²|ℎ|²+𝜀𝑘₂+𝜀𝑘₁. (26) On the other hand, considering that 𝜑(𝑢)+((𝛼−𝜀𝛾)/8)‖𝑢‖²≥−𝑘₁, fixing 𝜇₀>𝜌₀, and assuming that 𝐸(0)≤𝑅², then as 𝑡≥𝑡₀=𝑡₀(𝑅,𝜌₀)=(1/𝜀₀)log(𝑅/(𝜇²₀−𝜌²₀)), we have 𝐸(𝑡)≤𝜇²₀, (27) that is, |𝑣|²+𝛼−𝜀𝛾4‖𝑢‖²≤𝜇²₀. (28)
Take the inner products by Δ²𝑣 in both sides of (1); then make summation to get 12𝑑‖𝑣‖𝑑𝑡²+𝛼−𝜀𝛾Δ²𝑢²−𝜀‖𝑣‖²−𝛽+𝑀(𝑧(𝑡))+𝑁(𝑧̇(𝑡))Δ𝑢,Δ²𝑣+𝛾Δ²𝑣²+𝜀²Δ²𝑀(𝑧)(𝑧)̇𝑧̇+𝑔(𝑢)𝑢,𝑣+𝜀+𝑁,Δ²𝑣+𝑓(𝑣−𝜀𝑢),Δ²𝑣=ℎ,Δ²𝑣. (29) Considering the continuity of the functions 𝑀^′(⋅) and 𝑁^′(⋅), we have −𝛽+𝑀(𝑧(𝑡))+𝑁(𝑧̇(𝑡))Δ𝑢,Δ²𝑣≥−𝛽+𝐶₂𝜇²₀+𝐶₃𝜇²₀|Δ𝑢|Δ²𝑣≥−𝛽+𝐶₂𝜇²₀+𝐶₃𝜇²₀𝜇²₀𝛾−𝛾4Δ²𝑣², (30) where 𝐶₂,𝐶₃ are all positive constants. Also 𝜀²Δ²𝜀𝑢,𝑣≥−²𝜎²Δ²𝑢²−𝜀²4‖𝑣‖²,ℎ,Δ²𝑣=𝑑𝑑𝑡ℎ,Δ²𝑢+𝜀ℎ,Δ²𝑢. (31) In addition, with |𝑔^′(𝑢)|≤𝑘₀(1+|𝑢|^𝜌), there exists a constant 𝑘₃ such that |𝑔(𝑢)|_𝐿^∞≤𝑘₃,|𝑔^′(𝑢)|_𝐿^∞≤𝑘₃; so 𝑔(𝑢),Δ²𝑣=𝑑𝑔(𝑢)𝑑𝑡,Δ²𝑣−𝑔^′(𝑢)𝑢_𝑡,Δ²𝑢+𝜀𝑔(𝑢),Δ²𝑢≥𝑑𝑔(𝑢)𝑑𝑡,Δ²𝑢+𝜀𝑔(𝑢),Δ²𝑢−𝜀²8Δ²𝑢²−2𝑘²₃𝜇²₀𝜀². (32) Also by using Schwarz and Mean Value inequalities and Mean Value Theorem, we have 𝑓(𝑣−𝜀𝑢),Δ²𝑣≤𝛾4Δ²𝑣²+1𝛾𝑓^′(𝜉)²()≤𝛾𝑣−𝜀𝑢𝑑𝑥4Δ²𝑣²+𝛼²₂𝛾1+3𝜀²𝜇²₀, (33) where 𝜉 among 0 and 𝑣−𝜀𝑢. Set 𝑌₁(𝑡)=𝜀𝛼−𝜀²𝜀𝛾−²𝜎²−𝜀²8Δ²𝑢²+𝛾𝜆²2𝜀−𝜀−²4‖𝑣‖²+𝜀𝑔(𝑢),Δ²𝑢+𝜀ℎ,Δ²𝑢, (34) and write 𝑀=(𝛼²₂/𝛾)(1+3𝜀²)𝜇²₀+((𝛽+𝐶₂𝜇²₀+𝐶₃𝜇²₀)𝜇²₀)/𝛾+2𝑘²₃𝜇²₀/𝜀²; then (29) is transformed into 12𝑑𝐸𝑑𝑡¹(𝑡)+𝑌₁(𝑡)≤𝑀. (35) Here the function 𝐸₁(𝑡)=‖𝑣‖²+𝛼−𝜀𝛾Δ²𝑢²+2𝑔(𝑢),Δ²𝑢+2ℎ,Δ²𝑢 (36) is obtained by the energy function being changed slightly.
Let 0<𝜀≤min{𝜀₀,2𝛼/(𝛾+(2/𝜎²)+(1/4)),−3+9+2𝛾𝜆²,𝛼/(𝛾+(1/8))}, we have 𝑌₁(𝑡)≥(𝜀/2)𝐸₁(𝑡), and so 12𝑑𝐸𝑑𝑡¹(𝑡)+𝜀2𝐸₁(𝑡)≤𝑀,∀𝑡≥𝑡₀(𝐵). (37) Then an application of the Gronwall inequality leads to 𝐸₁(𝑡)≤𝐸₁(0)𝑒^{[−𝜀(𝑡−𝑡0)]}+2𝑀𝜀,∀𝑡≥𝑡₀(ß). (38) If 𝐵⊂𝐵_𝐸(0,𝜌), there exists a positive constant 𝑅₁>0 such that 𝐸₁(𝑡₀)≤𝑅²₁.
Putting 𝑡₁ satisfing 𝑡₁−𝑡₀>(1/𝜀)log𝑅²₁, then as 𝑡≥𝑡₁, we get 𝐸₁(𝑡)≤𝑅²₁𝑒^{−𝜀×(1/𝜀)log𝑅21}+2𝑀𝜀=1+2𝑀𝜀. (39) So 𝜀𝛼−𝜀𝛾−8Δ²𝑢²+‖𝑣‖²≤16𝜀|ℎ|²+16𝜀𝑘²₃|Ω|+1+2𝑀𝜀. (40) The global estimate (40) shows the existence of an absorbing set of 𝑆(𝑡).

4. The Existence of Global Attractor in Space 𝐸

The general theory [11] indicates that the continuous semigroup 𝑆(𝑡) defined on a Banach space 𝑋 has a global attractor which is connected when the following conditions are satisfied.(i) There exists a bounded absorbing set 𝐵⊂𝑋 such that for any bounded set 𝐵₀⊂𝑋, dist𝑆(𝑡)𝐵₀,𝐵⟶0,as𝑡⟶+∞. (41)(ii)𝑆(𝑡) is asymptotically compact; that is, for any bounded sequence {𝑢_𝑛} in 𝑋 and {𝑡_𝑛} tending to ∞, there exists a subsequence {𝑛^′} such that {𝑆(𝑡_𝑛′)𝑢_𝑛′} is convergent as 𝑛^′→∞.

Theorem 3. Under the assumptions of Theorem 1, the continuous semigroup 𝑆(𝑡) has a global attractor which is connected to 𝐸.

Proof. Let 𝑢,𝑣 be two solutions of Problems (1)–(3) in space 𝐶(𝑅⁺;𝐸) as shown above corresponding to the initial data (𝑢₀,𝑢₁) and (𝑣₀,𝑣₁) with |(𝑢₀,𝑢₁)|²_𝐸+|𝑣₀,𝑣₁|²_𝐸≤𝑅², respectively. Then 𝑤=𝑢−𝑣 satisfies 𝑤_𝑡𝑡+𝛼Δ²𝑤+𝛾Δ²𝑤_𝑡=𝑀||−𝛽Δ𝑤∇𝑢²Δ𝑢−𝑀|∇𝑣|²+𝑁Δ𝑣∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁∇𝑣∇𝑣_𝑡−𝑔(𝑢)(𝑣−𝑓𝑢𝑑𝑥Δ𝑣−𝑔)_𝑡−𝑓𝑣_𝑡,𝑤,𝑤_𝑡²_𝐸0≤𝐶𝜇²₀. (42)(43) Taking the inner products in both sides of (42) by 𝑤_𝑡, 𝐴𝑤, and 𝐴𝑤_𝑡, respectively, we have 12𝑑𝑤𝑑𝑡^𝑡2+𝛼|Δ𝑤|²+𝛽|∇𝑤|²+𝛾Δ𝑤_𝑡²=𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣,𝑤_𝑡+𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,𝑤_𝑡+𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,𝑤_𝑡,12𝑑𝛾Δ𝑑𝑡²𝑤²+2Δ𝑤,Δ𝑤_𝑡+𝛼Δ²𝑤²−𝛽Δ𝑤,Δ²𝑤+Δ𝑤_𝑡²=𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣,Δ²𝑤+𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,Δ²𝑤+𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,Δ²𝑤,12𝑑𝑑𝑡Δ𝑤_𝑡²+𝛼Δ²𝑤²+𝛾Δ²𝑤_𝑡²−𝛽Δ²𝑤,Δ𝑤_𝑡=𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣,Δ²𝑤_𝑡+𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,Δ²𝑤_𝑡+𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,Δ²𝑤_𝑡. (44)(45)(46) Equation (46)+𝑘×(45)+𝑘×(44) yields 12𝑑𝑑𝑡Δ𝑤_𝑡²+𝛼Δ²𝑤²+𝑘𝛾Δ²𝑤²+2𝑘Δ²𝑤,𝑤_𝑡+𝑘𝑤_𝑡²+𝑘𝛼|Δ𝑤|²+𝑘𝛽|∇𝑤|²+𝛾Δ²𝑤_𝑡²+𝑘𝛼Δ²𝑤²+𝑘Δ𝑤_𝑡²+𝑘𝛾Δ𝑤_𝑡²=𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,Δ²𝑤_𝑡+𝑘Δ²𝑤+𝑘𝑤_𝑡+𝑀|∇𝑢|²∇𝑢−𝑀|∇𝑣|²Δ𝑣+𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,Δ²𝑤_𝑡+𝑘𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣+𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,Δ²𝑤+𝑘𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣+𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,𝑤_𝑡+𝛽Δ𝑤,Δ²𝑤_𝑡+𝑘𝛽Δ𝑤,Δ²𝑤. (47) Consider that 𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣,Δ²𝑤_𝑡=𝑀^′𝜂₀|∇𝑢|²_ΩΔ𝑤Δ²𝑤_𝑡𝑑𝑥+𝑀^′𝜂₁×_Ω∇𝑤(Δ𝑢+∇𝑣)𝑑𝑥_ΩΔ𝑣Δ²𝑤_𝑡𝑑𝑥≤𝐶₂𝜇²₀|Δ𝑤|Δ²𝑤_𝑡+2𝐶₂𝜇²₀|∇𝑤|Δ²𝑤_𝑡𝐶≤2²𝜇²₀+2𝐶₂𝜇²₀/𝜎²𝛾|Δ𝑤|²+𝛾8Δ²𝑤_𝑡²;𝑘𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣,Δ²𝑤=𝑘𝑀^′𝜂₀|∇𝑢|²_ΩΔ𝑤Δ²𝑤𝑑𝑥+𝑘𝑀^′𝜂₁×_Ω∇𝑤(Δ𝑢+∇𝑣)𝑑𝑥_ΩΔ𝑣Δ²≤𝑤𝑑𝑥𝑘𝐶₂𝜇²₀|Δ𝑤|Δ²𝑤+2𝜇²₀𝜎|Δ𝑤|Δ²𝑤≤𝐶₂𝑘𝜇²₀(1+(2/𝜎))²||𝑘𝛼Δ𝑤²+𝑘𝛼8Δ²𝑤²;𝑘𝑀|∇𝑢|²Δ𝑢−𝑀|∇𝑣|²Δ𝑣,𝑤_𝑡=𝑘𝑀^′𝜂₀|∇𝑢|²_ΩΔ𝑤𝑤_𝑡𝑑𝑥+𝑘𝑀^′𝜂₁×_Ω∇𝑤(Δ𝑢+∇𝑣)𝑑𝑥_ΩΔ𝑣𝑤_𝑡≤𝑑𝑥𝑘𝐶₂𝜇²₀|Δ𝑤|𝑤_𝑡+2𝜇²₀𝜎|Δ𝑤|𝑤_𝑡≤𝐶₂𝑘𝜇²₀(1+(2/𝜎))²2|Δ𝑤|²+𝑤_𝑡², (48)(49)(50) where 𝜂₀ is among 0 and |∇𝑢|²,𝜂₁ is among |∇𝑢|² and |∇𝑣|², and 𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,Δ²𝑤_𝑡=𝑁^′𝜉_0Ω∇𝑢∇𝑢_𝑡𝑑𝑥_ΩΔ𝑤Δ²𝑤_𝑡𝑑𝑥+𝑁^′𝜉₁×_Ω∇𝑣∇𝑤_𝑡𝑑𝑥+_Ω∇𝑤∇𝑢_𝑡×𝑑𝑥_ΩΔ𝑣Δ²𝑤_𝑡𝑑𝑥≤𝐶₃𝜇²₀|Δ𝑤|Δ²𝑤_𝑡+𝜇₀𝑤_𝑡+𝜇₀|Δ𝑤|𝜇₀Δ²𝑤_𝑡≤4𝐶₃𝜇²₀²𝛾|Δ𝑤|²+𝛾8Δ²𝑤_𝑡²+4𝐶₃𝜇²₀²𝛾𝑤_𝑡²,𝑘𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,Δ²𝑤=𝑘𝑁^′𝜉_0Ω∇𝑢∇𝑢_𝑡𝑑𝑥_ΩΔ𝑤Δ²𝑤𝑑𝑥+𝑘𝑁^′𝜉₁×_Ω∇𝑣∇𝑤_𝑡𝑑𝑥+_Ω∇𝑤∇𝑢_𝑡𝑑𝑥×_ΩΔ𝑣Δ²≤𝑤𝑑𝑥𝑘𝐶₃𝜇²₀|Δ𝑤|Δ²𝑤+𝜇₀𝑤_𝑡+𝜇₀|Δ𝑤|𝜇₀Δ²𝑤≤4𝐶₃𝜇²₀²𝑘𝛼|Δ𝑤|²+𝑘𝛼8Δ²𝑤²+4𝐶₃𝜇²₀²𝑘𝛼𝑤_𝑡²,𝑘𝑁_Ω∇𝑢∇𝑢_𝑡𝑑𝑥Δ𝑢−𝑁_Ω∇𝑣∇𝑣_𝑡𝑑𝑥Δ𝑣,𝑤_𝑡=𝑘𝑁^′𝜉_0Ω∇𝑢∇𝑢_𝑡𝑑𝑥_ΩΔ𝑤𝑤_𝑡𝑑𝑥+𝑘𝑁^′𝜉₁×_Ω∇𝑣∇𝑤_𝑡𝑑𝑥+_Ω∇𝑤∇𝑢_𝑡𝑑𝑥×_ΩΔ𝑣𝑤_𝑡≤𝑑𝑥𝑘𝐶₃𝜇²₀|Δ𝑤|𝑤_𝑡+𝜇₀𝑤_𝑡+𝜇₀|Δ𝑤|𝜇₀𝑤_𝑡≤𝑘𝐶₃𝜇²₀|Δ𝑤|²+𝑘𝐶₃𝜇²₀𝑤_𝑡²+𝑘𝐶₃𝜇²₀𝑤_𝑡², (51)(52)(53) where 𝜉₀ is among 0 and ∫_Ω∇𝑢∇𝑢_𝑡𝑑𝑥, 𝜉₁ is among ∫_Ω∇𝑢∇𝑢_𝑡𝑑𝑥 and ∫_Ω∇𝑣∇𝑣_𝑡𝑑𝑥.
Also considering |𝑔(𝑢)−𝑔(𝑣)|≤𝑘₆(1+|𝑢|^𝜌+|𝑣|^𝜌)|𝑢−𝑣| for all 𝑢,𝑣∈𝑅, |𝑓(𝑢)−𝑓(𝑣)|≤𝑘₅(1+|𝑢|^𝑟+|𝑣|^𝑟)|𝑢−𝑣|, for all 𝑢,𝑣∈𝑅, and 𝜌/(2(𝜌+1))+1/(2(𝜌+1))+(1/2)=1, 𝑟/(2(𝑟+1))+1/(2(𝑟+1))+(1/2)=1, by Hölder inequality we have −𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,Δ²𝑤_𝑡≤𝑘_6Ω1+|𝑢|^𝜌+|𝑣|^𝜌|𝑤|Δ²𝑤_𝑡𝑑𝑥+𝑘_5Ω1+𝑢_𝑡^𝑟+𝑣_𝑡^𝑟𝑤_𝑡Δ²𝑤_𝑡𝑑𝑥≤𝑘_6Ω1+|𝑢|^𝜌+|𝑣|^{𝜌2(𝜌+1)/𝜌}𝑑𝑥^{𝜌/(2(𝜌+1))}×|𝑤|_2(𝜌+1)Δ²𝑤_𝑡+𝑘_5Ω1+𝑢_𝑡^𝑟+𝑣_𝑡^{𝑟2(𝑟+1)/𝑟}𝑑𝑥^{𝑟/(2(𝑟+1))}×𝑤_{𝑡2(𝑟+1)}Δ²𝑤_𝑡≤𝐶𝜇₀|∇𝑤|Δ²𝑤_𝑡+𝐶𝜇₀∇𝑤_𝑡Δ²𝑤_𝑡;−𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,𝑘Δ²𝑤≤𝑘𝑘_6Ω1+|𝑢|^𝜌+|𝑣|^𝜌|𝑤|Δ²𝑤+𝑑𝑥𝑘𝑘_5Ω1+𝑢_𝑡^𝑟+𝑣_𝑡^𝑟𝑤_𝑡Δ²𝑤≤𝑑𝑥𝑘𝑘_6Ω1+|𝑢|^𝜌+|𝑣|^{𝜌2(𝜌+1)/𝜌}𝑑𝑥^{𝜌/(2(𝜌+1))}×|𝑤|_2(𝜌+1)Δ²𝑤+𝑘𝑘_5Ω1+𝑢_𝑡^𝑟+𝑣_𝑡^{𝑟2(𝑟+1)/𝑟}𝑑𝑥^{𝑟/(2(𝑟+1))}×𝑤_{𝑡2(𝑟+1)}Δ²𝑤≤𝑘𝐶𝜇₀|𝑤|_2(𝜌+1)Δ²𝑤+𝑘𝐶𝜇₀𝑤_{𝑡2(𝑟+1)}Δ²𝑤≤𝑘𝐶𝜇₀|∇𝑤|Δ²𝑤+𝑘𝐶𝜇₀∇𝑤_𝑡Δ²𝑤;−𝑔(𝑢)−𝑔(𝑣)+𝑓𝑢_𝑡−𝑓𝑣_𝑡,𝑘𝑤_𝑡≤𝑘𝑘_6Ω1+|𝑢|^𝜌+|𝑣|^𝜌|𝑤|𝑤_𝑡+𝑑𝑥𝑘𝑘_5Ω1+𝑢_𝑡^𝑟+𝑣_𝑡^𝑟𝑤_𝑡𝑤_𝑡≤𝑑𝑥𝑘𝑘_6Ω1+|𝑢|^𝜌+|𝑣|^{𝜌2(𝜌+1)/𝜌}𝑑𝑥^{𝜌/(2(𝜌+1))}×|𝑤|_2(𝜌+1)𝑤_𝑡+𝑘𝑘_5Ω1+𝑢_𝑡^𝑟+𝑣_𝑡^{𝑟2(𝑟+1)/𝑟}𝑑𝑥^{𝑟/(2(𝑟+1))}×𝑤_{𝑡2(𝑟+1)}𝑤_𝑡≤𝑘𝐶𝜇₀|𝑤|_2(𝜌+1)𝑤_𝑡+𝑘𝐶𝜇₀𝑤_{𝑡2(𝑟+1)}𝑤_𝑡≤𝑘𝐶𝜇₀|∇𝑤|𝑤_𝑡+𝑘𝐶𝜇₀∇𝑤_𝑡𝑤_𝑡. (54)(55)(56) Setting 𝐸₂(𝑡)=Δ𝑤_𝑡²+𝛼Δ²𝑤²+𝑘𝛾Δ²𝑤²+2𝑘Δ²𝑤,𝑤_𝑡+𝑘𝑤_𝑡²+𝑘𝛼|Δ𝑤|²+𝑘𝛽|∇𝑤|²+𝛽|∇Δ𝑤|²,𝑌₂(𝑡)=𝛾2Δ²𝑤_𝑡²+𝑘𝛼2Δ²𝑤²+𝑘2Δ𝑤_𝑡²+𝑘𝛾2Δ𝑤_𝑡²+𝑘𝛽|∇Δ𝑤|², (57) then substituting (48)–(56) into (47), by Schwarz inequality and Young inequality, and taking 𝑘>(4𝐶²(𝜇₀))/𝛾𝜎² and 𝑘≥(8𝑘𝐶²(𝜇₀))/𝛼𝜎²𝛾, we have 12𝑑𝐸𝑑𝑡²(𝑡)+𝑌₂(𝑡)≤𝐶|Δ𝑤|²+𝑤_𝑡². (58) Again setting 𝜉=max{4/𝛼+4/𝑘+4𝛾/𝛼,2/𝑘,(4/𝜆²+4/𝛼𝜆⁴)(𝑘/𝑘),2/𝛾𝜆²+(2/𝛾𝜆²)(𝑘/𝑘)}, and considering that −2𝑘(Δ²𝑤,𝑤_𝑡)≥−𝑘|Δ²𝑤|²−𝑘|𝑤_𝑡|², we have 𝜉𝑌₂(𝑡)−𝐸₂(𝑡)≥0. On the one hand, from (58) we have 12𝑑𝐸₂(𝑡)+1𝑑𝑡𝜉𝐸₂(𝑡)≤𝐶|Δ𝑤|²+𝑤_𝑡². (59) Applying the Gronwall inequality to (59), we get 𝐸₂(𝑡)≤𝐸₂(0)𝑒^{−(2/𝜉)𝑡}+𝐶^𝑡₀𝑒^{−(2/𝜉)𝑡}‖𝑤(𝜏)‖²+𝑤_𝑡(𝜏)²𝑑𝜏. (60) On the other hand, with (2𝑘𝑤_𝑡,Δ²𝑤)≥−(𝑘𝛾/2)|Δ²𝑤|²−(4𝑘/𝛾)|𝑤_𝑡|² and setting 𝑘>4𝑘/𝛾, we get 𝐸₂(𝑡)≥Δ𝑤_𝑡²+𝛼Δ²𝑤². (61) Hence 𝑤,𝑤_𝑡²_𝐸≤𝐶𝐸₂(0)𝑒^{−2𝑡/𝜉}+𝐶^𝑡₀𝑒^{−(2/𝜉)𝑡}|Δ𝑤(𝜏)|²+𝑤_𝑡(𝜏)²𝑑𝜏. (62)
Now, let {(𝑢_0𝑚,𝑢_1𝑚)} be a bound sequence in 𝐵₀⊂𝐸, and {𝑢_𝑚(𝑡),𝑢_𝑚𝑡(𝑡)} the corresponding solutions of problems (1)–(3) in 𝐶(𝑅⁺,𝐸). We assume 𝑡_𝑛>𝑡_𝑚. Let 𝑇>0 and 𝑡_𝑛,𝑡_𝑚>𝑇. Then, applying estimate (62) to 𝑤^𝑚,𝑛=𝑢_𝑛(𝑡+𝑡_𝑛−𝑇)−𝑢_𝑚(𝑡+𝑡_𝑚−𝑇),𝑡≥0, we have 𝑤^𝑚,𝑛,𝑤_𝑡^𝑚,𝑛2_𝐸≤𝐶𝐶𝜇₀𝑒^{−(2/𝜉)𝑡}+𝐶×sup_{0≤𝑠≤𝑡}𝑢_𝑛𝑡_𝑛−𝑇+𝑠−𝑢_𝑚𝑡_𝑚,𝑢−𝑇+𝑠_𝑛𝑡𝑡_𝑛−𝑇+𝑠−𝑢_𝑚𝑡𝑡_𝑚−𝑇+𝑠²_𝐸0. (63) By taking 𝑡=𝑇 in the above, we have 𝑢_𝑛𝑡_𝑛−𝑢_𝑚𝑡_𝑚,𝑢_𝑛𝑡𝑡_𝑛−𝑢_𝑚𝑡𝑡_𝑚²_𝐸≤𝐶𝐶𝜇₀𝑒^{−(2/𝜉)𝑇}+𝐶𝜇₀×sup_{0≤𝑠≤𝑇}𝑢_𝑛𝑡_𝑛+𝑠−𝑢_𝑚𝑡_𝑚,𝑢+𝑠_𝑛𝑡𝑡_𝑛+𝑠−𝑢_𝑚𝑡𝑡_𝑚+𝑠²_𝐸0. (64) By Sobolev embedding Theorem, for any 𝑇>0, we can extract a subsequence {(𝑢_𝑛′,𝑢_𝑛′𝑡}which is convergent in 𝐶([0,𝑇];𝐸₀) for any 𝑇>0. For any 𝜀>0, we first fix 𝑇>0 such that 𝐶𝐶𝜇₀𝑒^{−(2/𝜉)𝑇}<𝜀2, (65) And, next, taking large 𝑚^′,𝑛^′, we have 𝐶𝜇₀sup_{0≤𝑠≤𝑇}𝑢_𝑛𝑡_𝑛+𝑠−𝑢_𝑚𝑡_𝑚,𝑢+𝑠_𝑛𝑡𝑡_𝑛+𝑠−𝑢_𝑚𝑡𝑡_𝑚+𝑠²_𝐸0≤𝜀2. (66) Then by (62) we have that 𝑢_𝑛′𝑡_𝑛′−𝑢_𝑚′𝑡_𝑚′,𝑢_𝑛′𝑡𝑡_𝑛′−𝑢_𝑚′𝑡𝑡_𝑚′²_𝐸≤𝜀. (67) We conclude that 𝑆(𝑡) is asymptotically compact on 𝐸. The theorem is now proved.

Acknowledgments

This work is partially supported by the Natural Science Foundation of China (11172194) and Shanxi Province (2011021002-2 and 2010011008). The authors also wish to give their thanks to the referees for their comments to improve the presentation of this paper.