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)×H01(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
(1)utt+αΔ2u+γΔ2ut-(β+M(∫Ω|∇u|2dx)+N(∫Ω∇u∇utdx))Δu+g(u)+f(ut)=h(x),inΩ×R+,(2)u=Δu=0on∂Ω×R+,(3)u(x,0)=u0(x),ut(x,0)=u1(x)inΩ,
where α,γ, and β are all positive constants, Ω is a bounded domain of RN 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
(4)utt+uxxxx-(α+β∫0l|ux(s,t)|2ds)uxx=0
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 H02(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(ut)=ut, 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 ut or nonlinear damping f(ut) 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)×H01(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)×H01(Ω)∩H2(Ω)), in Section 3 we give the boundedness of solutions in phase space D(Δ2)×H01(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)×H01(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
(5)f(ut)=|ut|rut,g(u)=|u|ρu
with
(6)0<ρ,r≤2N-2ifN≥3,ρ,r>0ifN=1,2.

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

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
(8)|f(u)-f(v)|≤k5(1+|u|r+|v|r)|u-v|,∀u,v∈R,|g(u)-g(v)|≤k6(1+|u|ρ+|v|ρ)|u-v|,∀u,v∈R.

In addition, nonlinear function g(·) also satisfies
(9)φ(u)+α-εγ8∥u∥2≥-k1,∫Ωg(u)udx-C1φ(u)+α4∥u∥2≥-k2,
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=H01(Ω)∩H2(Ω), with the usual inner products and norms as follows, respectively:
(10)(u,v)=∫Ωuvdx,|u|=(u,u)1/2,∀u,v∈L2(Ω),(Δu,Δv)=∫ΩΔuΔvdx,∥u∥=(Δu,Δu)1/2,∀u,v∈H01(Ω)∩H2(Ω).

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=H01(Ω)∩H2(Ω)×L2(Ω) and E=D(Δ2)×H2(Ω)∩H01(Ω) with the inner products and norms as follows, respectively:
(11)(y1,y2)E0=(Δu1,Δu2)+(v1,v2),|y|E0=(y,y)E01/2,∀yi=(ui,vi)T,y=(u,v)T∈E0,i=1,2,(y1,y2)E=(Δ2u1,Δ2u2)+(Δv1,Δv2),|y|E=(y,y)E1/2,∀yi=(ui,vi)T,y=(u,v)T∈E,i=1,2.

Note that assumption (6) implies that H01(Ω)∩H2(Ω)↪H01(Ω)↪L2(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
(12)∥u∥2≥σ|u|2,∀u∈V,|Δ2u|2≥λ∥v∥2,∀u∈D(Δ2).
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,ut)∈C([0,T];D(Δ2))×C([0,T];H01(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
(13){S(t),t≥0}:{u0,u1}⟶{u(t),ut(t)}.
It maps E into itself, and it enjoys the usual semigroup properties as follows:
(14)S(0)=I,S(t+τ)=S(t)S(τ),∀t≥0.
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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M88"><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:math></inline-formula>

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=ut+εu with both sides of (1) and then making summation, we have
(15)12ddtE(t)-ε|v|2+ε(α-εγ)∥u∥2+εβ|∇u|2+γ|v|2+ε2(u,v)+εM^(z)+N(z˙)z˙+(g(u),v)+(f(ut),v)=(h,v),
where M^(z)=∫0zM(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
(16)E(t)=|v|2+(α-εγ)∥u∥2+β|∇u|2+M^(z)+|∇u|4.
Considering the assumption ∫Ωug(u)dx-C1φ(u)+(α/4)∥u∥2≥-k2, we have
(17)(g(u),v)=ddtφ(u)+ε(g(u),u)≥ddtφ(u)+εC1φ(u)-εα4∥u∥2-εk2.
With |ε2(u,v)|≤(ε2/σ2)∥u∥2+(ε2/4)|v|2, we have
(18)-ε|v|2+ε(α-εγ)∥u∥2+εβ|∇u|2+γ|v|2+ε2(u,v)≥(εα-ε2γ-ε2σ2)∥u∥2+(γλ2-ε-ε24)|v|2+εβ|∇u|2.
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
(19)(f(ut),v)=∫f′(ξ)v2dx-ε∫f′(ξ)uvdx≥(α1-εα22)|v|2-εα22σ2∥u∥2,
where ξ among 0 and v-εu. Set
(20)E~(t)=|v|2+(α-εγ)∥u∥2+β|∇u|2+M^(z)+|∇u|4+2φ(u)+2k1.
Consider
(21)Y(t)=(εα-ε2γ-ε2σ2-εα4-εα22σ2)∥u∥2+(γλ2-ε-ε24+α1-γσ24-εα22)|v|2+εβ|∇u|2+εM^(z)+N(z˙)z˙+εC1φ(u)+εk1.
So (15) is transformed into
(22)12ddtE~(t)+Y(t)≤1γσ2|h|2+εk2+εk1.
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
(23)2εY(t)-E~(t)>0.
Substituting (23) into (22), we have
(24)12ddtE~(t)+ε2E~(t)≤1γσ2|h|2+εk2+εk1.
On the one hand, applying the Gronwall inequality to (24), we get
(25)E~(t)≤E~(0)e-εt+2ε(1γσ2|h|2+εk2+εk1),t≥0.
Note that ∥u(0)∥ and |ut(0)| are bounded; then there exists a positive constant R>0 such that E~(0)≤R2 is bounded; so
(26)limsupt→∞E~(t)≤ρ02=2ε(1γσ2|h|2+εk2+εk1).
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/(μ02-ρ02)), we have
(27)E~(t)≤μ02,
that is,
(28)|v|2+α-εγ4∥u∥2≤μ02.

Take the inner products by Δ2v in both sides of (1); then make summation to get
(29)12ddt(∥v∥2+(α-εγ)|Δ2u|2)-ε∥v∥2-(β+M(z(t))+N(z˙(t)))(Δu,Δ2v)+γ|Δ2v|2+ε2(Δ2u,v)+εM^(z)+N(z˙)z˙+(g(u),Δ2v)+(f(v-εu),Δ2v)=(h,Δ2v).
Considering the continuity of the functions M′(·) and N′(·), we have
(30)-(β+M(z(t))+N(z˙(t)))(Δu,Δ2v)≥-(β+C2μ02+C3μ02)|Δu||Δ2v|≥-(β+C2μ02+C3μ02)μ02γ-γ4|Δ2v|2,
where C2,C3 are all positive constants. Also
(31)ε2(Δ2u,v)≥-ε2σ2|Δ2u|2-ε24∥v∥2,(h,Δ2v)=ddt(h,Δ2u)+ε(h,Δ2u).
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
(32)(g(u),Δ2v)=ddt(g(u),Δ2v)-(g′(u)ut,Δ2u)+ε(g(u),Δ2u)≥ddt(g(u),Δ2u)+ε(g(u),Δ2u)-ε28|Δ2u|2-2k32μ02ε2.
Also by using Schwarz and Mean Value inequalities and Mean Value Theorem, we have
(33)(f(v-εu),Δ2v)≤γ4|Δ2v|2+1γ∫(f′(ξ))2(v-εu)dx≤γ4|Δ2v|2+α22γ(1+3ε2)μ02,
where ξ among 0 and v-εu. Set
(34)Y1(t)=(εα-ε2γ-ε2σ2-ε28)|Δ2u|2+(γλ22-ε-ε24)∥v∥2+ε(g(u),Δ2u)+ε(h,Δ2u),
and write M=(α22/γ)(1+3ε2)μ02+((β+C2μ02+C3μ02)μ02)/γ+2k32μ02/ε2; then (29) is transformed into
(35)12ddtE1(t)+Y1(t)≤M.
Here the function
(36)E1(t)=∥v∥2+(α-εγ)|Δ2u|2+2(g(u),Δ2u)+2(h,Δ2u)
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
(37)12ddtE1(t)+ε2E1(t)≤M,∀t≥t0(B).
Then an application of the Gronwall inequality leads to
(38)E1(t)≤E1(0)e[-ε(t-t0)]+2Mε,∀t≥t0(ß).
If B⊂BE(0,ρ), there exists a positive constant R1>0 such that E1(t0)≤R12.

Putting t1 satisfing t1-t0>(1/ε)logR12, then as t≥t1, we get
(39)E1(t)≤R12e-ε×(1/ε)logR12+2Mε=1+2Mε.
So
(40)(α-εγ-ε8)|Δ2u|2+∥v∥2≤16ε|h|2+16εk32|Ω|+1+2Mε.
The global estimate (40) shows the existence of an absorbing set of S(t).

4. The Existence of Global Attractor in Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M164"><mml:mrow><mml:mi>E</mml:mi></mml:mrow></mml:math></inline-formula>

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.

There exists a bounded absorbing set B⊂X such that for any bounded set B0⊂X,
(41)dist(S(t)B0,B)⟶0,ast⟶+∞.

S(t) is asymptotically compact; that is, for any bounded sequence {un} in X and {tn} tending to ∞, there exists a subsequence {n′} such that {S(tn′)un′} 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)|E2+|v0,v1|E2≤R2, respectively. Then w=u-v satisfies
(42)wtt+αΔ2w+γΔ2wt-βΔw=(M(|∇u|2)Δu-M(|∇v|2)Δv)+(N(∫∇u∇utdx)Δu-N(∫∇v∇vtdx)Δv)-(g(u)-g(v))-(f(ut)-f(vt)),(43)|(w,wt)|E02≤C(μ02).
Taking the inner products in both sides of (42) by wt, Aw, and Awt, respectively, we have
(44)12ddt(|wt|2+α|Δw|2+β|∇w|2)+γ|Δwt|2=(M(|∇u|2)Δu-M(|∇v|2)Δv,wt)+(N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,wt)+(g(u)-g(v)+f(ut)-f(vt),wt),(45)12ddt(γ|Δ2w|2+2(Δw,Δwt))+α|Δ2w|2-β(Δw,Δ2w)+|Δwt|2=(M(|∇u|2)Δu-M(|∇v|2)Δv,Δ2w)+(N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,Δ2w)+(g(u)-g(v)+f(ut)-f(vt),Δ2w),(46)12ddt(|Δwt|2+α|Δ2w|2)+γ|Δ2wt|2-β(Δ2w,Δwt)=(M(|∇u|2)Δu-M(|∇v|2)Δv,Δ2wt)+(N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,Δ2wt)+(g(u)-g(v)+f(ut)-f(vt),Δ2wt).
Equation (46)+k~×(45)+k~~×(44) yields
(47)12ddt(k~~β|∇w|2|Δwt|2+α|Δ2w|2+k~γ|Δ2w|2+2k~(Δ2w,wt)+k~~|wt|2+k~~α|Δw|2+k~~β|∇w|2)+γ|Δ2wt|2+k~α|Δ2w|2+k~|Δwt|2+k~~γ|Δwt|2=(g(u)-g(v)+f(ut)-f(vt),Δ2wt+k~Δ2w+k~~wt)+((∫Ω∇v∇vtdx)M(|∇u|2)∇u-M(|∇v|2)Δv+N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,Δ2wt)+k~((∫Ω∇v∇vtdx)M(|∇u|2)Δu-M(|∇v|2)Δv+N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,Δ2w)+k~~((∫Ω∇v∇vtdx)M(|∇u|2)Δu-M(|∇v|2)Δv+N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,wt)+(βΔw,Δ2wt)+k~β(Δw,Δ2w).
Consider that
(48)(M(|∇u|2)Δu-M(|∇v|2)Δv,Δ2wt)=M′(η0)|∇u|2∫ΩΔwΔ2wtdx+M′(η1)×∫Ω∇w(Δu+∇v)dx∫ΩΔvΔ2wtdx≤C2μ02|Δw||Δ2wt|+2C2μ02|∇w||Δ2wt|≤2(C2μ02+(2C2μ02/σ))2γ|Δw|2+γ8|Δ2wt|2;(49)k~(M(|∇u|2)Δu-M(|∇v|2)Δv,Δ2w)=k~M′(η0)|∇u|2∫ΩΔwΔ2wdx+k~M′(η1)×∫Ω∇w(Δu+∇v)dx∫ΩΔvΔ2wdx≤k~C2(μ02|Δw||Δ2w|+2μ02σ|Δw||Δ2w|)≤[C2k~μ02(1+(2/σ))]2k~α|Δw|2+k~α8|Δ2w|2;(50)k~~(M(|∇u|2)Δu-M(|∇v|2)Δv,wt)=k~~M′(η0)|∇u|2∫ΩΔwwtdx+k~~M′(η1)×∫Ω∇w(Δu+∇v)dx∫ΩΔvwtdx≤k~~C2(μ02|Δw||wt|+2μ02σ|Δw||wt|)≤[C2k~~μ02(1+(2/σ))]22|Δw|2+|wt|2,
where η0 is among 0 and |∇u|2,η1 is among |∇u|2 and |∇v|2, and
(51)(N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,Δ2wt)=N′(ξ0)∫Ω∇u∇utdx∫ΩΔwΔ2wtdx+N′(ξ1)×(∫Ω∇v∇wtdx+∫Ω∇w∇utdx)×∫ΩΔvΔ2wtdx≤C3[μ02|Δw||Δ2wt|+(μ0|wt|+μ0|Δw|)μ0|Δ2wt|]≤(4C3μ02)2γ|Δw|2+γ8|Δ2wt|2+(4C3μ02)2γ|wt|2,(52)k~(N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,Δ2w)=k~N′(ξ0)∫Ω∇u∇utdx∫ΩΔwΔ2wdx+k~N′(ξ1)×(∫Ω∇v∇wtdx+∫Ω∇w∇utdx)×∫ΩΔvΔ2wdx≤k~C3[μ02|Δw||Δ2w|+(μ0|wt|+μ0|Δw|)μ0|Δ2w|]≤(4C3μ02)2k~α|Δw|2+k~α8|Δ2w|2+(4C3μ02)2k~α|wt|2,(53)k~~(N(∫Ω∇u∇utdx)Δu-N(∫Ω∇v∇vtdx)Δv,wt)=k~~N′(ξ0)∫Ω∇u∇utdx∫ΩΔwwtdx+k~~N′(ξ1)×(∫Ω∇v∇wtdx+∫Ω∇w∇utdx)×∫ΩΔvwtdx≤k~~C3[μ02|Δw||wt|+(μ0|wt|+μ0|Δw|)μ0|wt|]≤k~~C3μ02|Δw|2+k~~C3μ02|wt|2+k~~C3μ02|wt|2,
where ξ0 is among 0 and ∫Ω∇u∇utdx, ξ1 is among ∫Ω∇u∇utdx and ∫Ω∇v∇vtdx.

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
(54)-(g(u)-g(v)+f(ut)-f(vt),Δ2wt)≤k6∫Ω(1+|u|ρ+|v|ρ)|w||Δ2wt|dx+k5∫Ω(1+|ut|r+|vt|r)|wt||Δ2wt|dx≤k6[∫Ω(1+|u|ρ+|v|ρ)2(ρ+1)/ρdx]ρ/(2(ρ+1))×|w|2(ρ+1)|Δ2wt|+k5[∫Ω(1+|ut|r+|vt|r)2(r+1)/rdx]r/(2(r+1))×|wt|2(r+1)|Δ2wt|≤C(μ0)|∇w||Δ2wt|+C(μ0)|∇wt||Δ2wt|;(55)-(g(u)-g(v)+f(ut)-f(vt),k~Δ2w)≤k~k6∫Ω(1+|u|ρ+|v|ρ)|w||Δ2w|dx+k~k5∫Ω(1+|ut|r+|vt|r)|wt||Δ2w|dx≤k~k6[∫Ω(1+|u|ρ+|v|ρ)2(ρ+1)/ρdx]ρ/(2(ρ+1))×|w|2(ρ+1)|Δ2w|+k~k5[∫Ω(1+|ut|r+|vt|r)2(r+1)/rdx]r/(2(r+1))×|wt|2(r+1)|Δ2w|≤k~C(μ0)|w|2(ρ+1)|Δ2w|+k~C(μ0)|wt|2(r+1)|Δ2w|≤k~C(μ0)|∇w||Δ2w|+k~C(μ0)|∇wt||Δ2w|;(56)-(g(u)-g(v)+f(ut)-f(vt),k~~wt)≤k~~k6∫Ω(1+|u|ρ+|v|ρ)|w||wt|dx+k~~k5∫Ω(1+|ut|r+|vt|r)|wt||wt|dx≤k~~k6[∫Ω(1+|u|ρ+|v|ρ)2(ρ+1)/ρdx]ρ/(2(ρ+1))×|w|2(ρ+1)|wt|+k~~k5[∫Ω(1+|ut|r+|vt|r)2(r+1)/rdx]r/(2(r+1))×|wt|2(r+1)|wt|≤k~~C(μ0)|w|2(ρ+1)|wt|+k~~C(μ0)|wt|2(r+1)|wt|≤k~~C(μ0)|∇w||wt|+k~~C(μ0)|∇wt||wt|.
Setting
(57)E2(t)=|Δwt|2+α|Δ2w|2+k~γ|Δ2w|2+2k~(Δ2w,wt)+k~~|wt|2+k~~α|Δw|2+k~~β|∇w|2+β|∇Δw|2,Y2(t)=γ2|Δ2wt|2+k~α2|Δ2w|2+k~2|Δwt|2+k~~γ2|Δwt|2+k~β|∇Δw|2,
then substituting (48)–(56) into (47), by Schwarz inequality and Young inequality, and taking k~>(4C2(μ0))/γσ2 and k~~≥(8k~C2(μ0))/ασ2γ, we have
(58)12ddtE2(t)+Y2(t)≤C(|Δw|2+|wt|2).
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,wt)≥-k~|Δ2w|2-k~|wt|2, we have ξY2(t)-E2(t)≥0. On the one hand, from (58) we have
(59)12dE2(t)dt+1ξE2(t)≤C(|Δw|2+|wt|2).
Applying the Gronwall inequality to (59), we get
(60)E2(t)≤E2(0)e-(2/ξ)t+C∫0te-(2/ξ)t(∥w(τ)∥2+|wt(τ)|2)dτ.
On the other hand, with (2k~wt,Δ2w)≥-(k~γ/2)|Δ2w|2-(4k~/γ)|wt|2 and setting k~~>4k~/γ, we get
(61)E2(t)≥|Δwt|2+α|Δ2w|2.
Hence
(62)|w,wt|E2≤CE2(0)e-2t/ξ+C∫0te-(2/ξ)t(|Δw(τ)|2+|wt(τ)|2)dτ.

Now, let {(u0m,u1m)} be a bound sequence in B0⊂E, and {um(t),umt(t)} the corresponding solutions of problems (1)–(3) in C(R+,E). We assume tn>tm. Let T>0 and tn,tm>T. Then, applying estimate (62) to wm,n=un(t+tn-T)-um(t+tm-T),t≥0, we have
(63)|(wm,n,wtm,n)E2≤CC(μ0)e-(2/ξ)t+C×sup0≤s≤t|(un(tn-T+s)-um(tm-T+s)),(unt(tn-T+s)-umt(tm-T+s))|E02.
By taking t=T in the above, we have
(64)|(un(tn)-um(tm),unt(tn)-umt(tm))|E2≤CC(μ0)e-(2/ξ)T+C(μ0)×sup0≤s≤T|(un(tn+s)-um(tm+s)),(unt(tn+s)-umt(tm+s))|E02.
By Sobolev embedding Theorem, for any T>0, we can extract a subsequence {(un′,un′t}which is convergent in C([0,T];E0) for any T>0. For any ε>0, we first fix T>0 such that
(65)CC(μ0)e-(2/ξ)T<ε2,
And, next, taking large m′,n′, we have
(66)C(μ0)sup0≤s≤T|(un(tn+s)-um(tm+s),unt(tn+s)-umt(tm+s))(un(tn+s)-um(tm+s))|E02≤ε2.
Then by (62) we have that
(67)|(un′(tn′)-um′(tm′),un′t(tn′)-um′t(tm′))|E2≤ε.
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.

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