# Attractor of Beam Equation with Structural Damping under Nonlinear Boundary Conditions

Authors: Danxia Wang; Jianwen Zhang; Yinzhu Wang; Sufang Zhang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Simultaneously, considering the viscous effect of material, damping of medium, and rotational inertia, we study a kind of more general Kirchhoff-type extensible beam equationutt-uxxtt+uxxxx-σ(∫0l‍(ux)2dx)uxx-ϕ(∫0l‍(ux)2dx)uxxt=q(x),in[0,L]×R+ with the structural damping and the rotational inertia term. Little  attention is paid to the  longtime behavior of the beam equation under nonlinear boundary conditions. In this paper, under nonlinear boundary conditions, we prove not only the existence and uniqueness of global solutions by prior estimates combined with some inequality skills, but also the existence of a global attractor by the existence of an absorbing set and asymptotic compactness of corresponding solution semigroup. In addition, the same results also can be proved under the other nonlinear boundary conditions.


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## Body

## 1. Introduction

In this paper, we will consider a kind of more general Kirchhoff-type beam equation. The physical origin of the problem lies in the theory of vibrations of an extensible beam of lengthL; moreover, during vibration, the elements of a beam not only perform a translatory motion but also rotate.A mathematical model for this problem is an initial boundary value problem for the nonlinear Kirchhoff-type beam equation(1)utt-uxxtt+uxxxx-σ∫0L‍ux2dxuxx-ϕ∫0L‍ux2dxuxxt=qx,x,t∈0,L×R+.
We assume the nonlinear boundary conditions
(2)u0,t=uxL,t=uxx0,t=0,(3)uxxxL,t=fuL,t+gutL,t
and the initial conditions
(4)u(x,0)=u0(x),ut(x,0)=u1(x),x∈(0,L).
Here the unknown function u(x,t) is the elevation of the surface of beam, u0(x) and u1(x) are the given initial value functions, and the subscript t and x denote derivative with respect to t and x, respectively. -uxxtt expresses the rotational inertia, and nonlinear terms σ(∫0L‍(ux)2dx)uxx and ϕ(∫0L‍(ux)2dx)uxxt represent the extensibility effects and the structural damping, respectively. q(x) is a static load. Moreover the assumptions on nonlinear functions σ(·), ϕ(·), g(·), and f(·) and the external force function q(x) will be specified later.In (1), when the structural damping term and the rotational inertia term are absent, (1) is a model for vibrations of tensible beam. This was proposed by Woinowsky-Krieger [1] in the form
(5)utt+uxxxx-α+β∫0L‍ux2dxuxx=0.
One of the first mathematical analyses for equation
(6)utt+uxxxx-M∫0L‍ux2dxuxx=0
was done by Ball [2] which was later extended to an abstract setting by defining a linear operator A by Medeiros [3]. In [4], Patcheu obtained the decay of the energy for above equation when a nonlinear damping g(ut) was effective in Ω. In addition, the attractor on extensible beams with null boundary conditions was considered by several authors. We quote, for instance, [5–7], and so on. But the longtime behavior of the beam equation with nonlinear boundary conditions was paid little attention. We also refer the reader to a few works. One of the first studies in this direction was done by Pazoto and Menzala [8], where stabilization of a thermoelastic extensible beam was considered. Motivated by the result, Ma proved the existence of global solutions and the existence of a global attractor in [9] and [10], respectively, for the Kirchhoff-type beam equation
(7)utt+uxxxx-M∫0L‍ux2dxuxx=0,
with the absence of the structural damping and the rotational inertia, subjected to the nonlinear boundary conditions
(8)u(0,t)=ux(0,t)=uxx(L,t)=0,uxxx(L,t)-M∫0L‍ux2dxuxL,t=fuL,t+gutL,t.In the following, we mentioned some results on longtime behavior of beam equation with the rotational inertia term. Under null boundary conditions, Geredeli and Lasiecka [11] considered the existence of a compact attractor of beam
(9)utt-αΔutt+Δ2u+dxgut-αdiv⁡dxg∇ut-Fu=px
with a rotational inertia term. Under nonlinear boundary conditions
(10)u=0,Δu=-g∂ut∂υ,
Ji and Lasiecka [12] considered the semilinear Kirchhoff equation
(11)utt-γΔutt+Δ2u+fu=0
with rotational inertia, and they showed that the above problem is uniformly stabilized with uniform energy decay rates.In addition, we also mentioned some results on longtime behavior of the equation with the structural damping term. Chueshov [13] studied the global attractor with a structural damping of the form σ(∇u2)(-Δ)θut with 1/2≤θ≤1. Chueshov [14] and Yang et al. [15] considered the global attractor for the Kirchhoff-type equation
(12)utt-σ∇u2Δut-ϕ∇u2Δu+f(u)=h(x)
with structural damping under null boundary conditions, respectively.On (1) under the following other nonlinear boundary conditions
(13)uL,t=ux(0,t)=uxx(L,t)=0,uxxx0,t=fu0,t+gut0,t
we also can get the same on the existence of global solutions and the existence of global attractor.Our fundamental assumptions onσ(·), ϕ(·), g(·), f(·), and q(x) are given as follows.Assumption 1.
We assume thatσ(·), ϕ(·)∈C1(R) are all nondecreasing and satisfy
(14)σzz≥σ^z≥0,∀z≥0,
where σ^(z)=∫0z‍σ(s)ds≥0. Moreover
(15)σ0=0,ϕ0=0,(16)σs≥ϕs≥α+βsγα,β>0,γ≥1,∀s∈R+.Assumption 2.
The functiong(·):R→R is of class C1(R) and satisfies g(0)=0, and there exist constants k1,k2 and r≥0 such that
(17)gu-gvu-v≥k1u-v2,∀u,v∈R,(18)gu-gv≤k21+ur+vru-v,∀u,v∈R.Assumption 3.
The functionf(·):R→R is of class C1(R) and satisfies f(0)=0, and there exist constants k and ϱ≥0 such that
(19)fu-fv≤k1+uϱ+vϱu-v,∀u,v∈R,(20)-L0≤f^u≤fuu+L1,
where f^(z)=∫0z‍f(s)ds≥0.Assumption 4.
Considerq(x)∈L2(0,L).Under the above assumptions, we prove the existence of global solutions and the existence of a global attractor of extensible beam equation system (1)–(4). And the paper is organized as follows. In Section 2, we introduce some Sobolev spaces. In Section 3, we discuss the existence of global strong and weak solutions. In Section 4, we establish the result of the existence of a global attractor in V×U.


## 2. Preliminaries

Our analysis is based on the following Sobolev spaces. Let(21)U=u∈H10,L;u0=0,V=u∈H20,L;u0=uxL=0,W=u∈H40,L∩V;uxx=0.
Motivated by the boundary condition (3) we assume, for regular solutions, that data (u0,u1) satisfies the following compatibility condition:
(22)uxxx0L=fu0L+gu1L.
Then for regular solutions we consider the phase space
(23)H1=u0,u1∈W×W;satisfiescondition22.
In the case of weak solutions we consider the phase space
(24)H0=V×U=H1¯,
which guarantees that, for regular data, the nonlinear condition (22) holds. In H0 we adopt the norm defined by
(25)u,vH02=u,vV×U2=uxx2+vx2.


## 3. The Existence of Global Solutions

Firstly, using the classical Galerkin method, we can establish the existence and uniqueness of regular solution to problem (1)–(4). We state it as follows.Theorem 1.
Assume Assumptions1–4 and (22)-(23) hold, for any initial data (u0,u1)∈H1; then problem (1)–(4) has a unique regular solution u(t) with
(26)u∈Lloc⁡∞R+,W∩C00,∞;V∩C10,∞;U.
Moreover,
(27)ut2+uxt2+uxx2+σ^ux2≤M1,
where M1>0 depends on the initial data and q, but not on t>0.Proof.
Let us consider the variational problem associated with (1)–(4): find u(t)∈W such that
(28)∫0L‍uttωdx+∫0L‍uxttωxdx+∫0L‍uxxωxxdx+σux2∫0L‍uxωxdx+ϕux2∫0L‍uxtωxdx+f(u(L,t))ω(L)+g(ut(L,t))ω(L)=∫0L‍qxωdx
for all ω∈V. This is done with the Galerkin approximation methods which is standard. Here we denote the approximate solution by um(t). We can get the theorem by proving the existence of approximation solution, the estimate of approximation solution, convergence, uniqueness, and u∈C0([0,∞);V)∩C1([0,∞);U). In the following we give the estimates of approximation solution, the proof of uniqueness of solution, and the proof of u∈C0([0,∞);V)∩C1([0,∞);U).Estimate  1. In approximate equation of (28), putting ω=utm(t) and considering σ^(z)=∫0z‍σ(s)ds, f^(z)=∫0z‍f(s)ds, using Schwarz inequality, and then integrating from 0 to t<tm, we see that
(29)utm2+uxtm2+uxxm2+σ^(uxm2)+f^uml,t+2∫0t‍ϕuxm2uxtm2ds+2∫0t‍gutml,tutml,tds≤∫0t‍q(x)2ds+∫0t‍utm2ds+utm(0)2+uxtm02+uxxm02+σ^uxm02+f^uml,0.
Taking into account the assumptions ∫0t‍g(utm(l,t))utm(l,t)ds≥0, ϕ(·)≥0, and f^,σ^≥0 of f, g, σ, and ϕ, we see that there exists M1>0 such that
(30)utm2+uxtm2+uxxm2+σ^uxm2≤M1
for all t∈[0,T] and for all m∈N.Estimate  2. In approximate equation of (28), integrating by parts with ω=uttm(0) and t=0 and considering the compatibility condition (22) and then using Schwarz inequality and the mean value inequality, we see that there exists M2>0 such that
(31)uttm02+uxttm02≤M2
for all t∈[0,T] and for all m∈N.Estimate 3. Let us fix t,ξ>0 such that ξ<T-t. Taking the difference of approximate equation of (28) with t=t+ξ and t=t and replacing ω by utm(t+ξ)-utm(t), we can find a constant M3>0, depending only on T, such that
(32)uttm2+uxttm2+uxxtm2≤M3,∀m∈N,∀t∈[0,T].Uniqueness. Let u,v be two solutions of (1)–(4) with the same initial data. Then writing z=u-v and taking the difference (28) with u=u and u=v and replacing ω by zt and then using mean value theorem and the Young inequalities combined with Estimates 1 and 3, we deduce that, for some constant C>0,
(33)ddtztm2+zxtm2+zxxm2≤Cztm2+zxtm2+zxxm2,∀t∈0,T.
Then from Gronwall’s lemma we see that u=v.The Proof of
u
∈
C
0
(
[
0
,
∞
)
;
V
)
∩
C
1
(
[
0
,
∞
)
;
U
)
. Since uxx,uxxt∈L2(0,∞;L2(0,L)), we get u(x,t)∈C0([0,∞);V). Similarly, u(x,t)∈C1([0,∞);U). The proof of Theorem 1 is completed.Theorem 2.
Assume the assumptions of Theorem1 and (24) hold; if the initial data (u0,u1)∈H0, then there exists a unique weak solution u(t) of problem (1)–(4) which depends continuously on initial data with respect to the norm of  V×U.Proof.
Let us consider{u0,u1}∈W×W¯=V×L2, and since W×W is dense in V×U, there exists {uμ0,uμ1}⊂W×W, such that
(34)uμ0⟶u0,inV;uμ1⟶u1,inU.
We observe that, for each μ∈N, there exists uμ, smooth solution of the initial boundary value problem (1)–(4) which satisfies
(35)uttμ-uxxttμ+uxxxxμ-σ∫0L‍uxμ2dxuxxμ-ϕ∫0L‍uxμ2dxuxxtμ=qx,x,t∈0,L×R+,uμx,0=uμ0x,utμx,0=uμ1x,x∈0,L,uμ0,t=uxμL,t=uxxμ0,t=0,t∈0,∞,uxxxμL,t=fuμL,t+gutμL,t,t∈0,∞.
Considering the arguments used in the estimate of the existence of solution, we obtain
(36)utμ2+uxtμ2+uxxμ2≤C0,
where C0 is a positive constant independent of μ∈N.
DefiningZμ,σ=uμ-uσ:μ, σ∈N where uμ and uσ are regular solutions of (35), following the steps already used in the uniqueness of regular solution for (1)–(4) and considering the convergence given in (34)  (uμ0→u0inVanduμ1→u1inU), we deduce that there exists u such that
(37)uμ⟶ustronglyinC0,T;V,utμ⟶utstronglyinC0,T;U.
From the above convergence, we can pass to the limit using standard arguments in order to obtain
(38)utt-uxxtt+uxxxx-σ∫0L‍ux2dxuxx-ϕ∫0L‍ux2dxuxxt=qx,(x,t)∈(0,L)×R+,u(x,0)=u0(x),ut(x,0)=u1(x),x∈(0,L).
Theorem 2 is proved.Remark 3.
Theorem2 implies that problem (1)–(4) defines a nonlinear C0-semigroup S(t) on H0. Indeed, let us set S(t)(u0,u1)=(u(t),ut(t)), where u is the unique solution corresponding to initial data (u0,u1)∈H0. Moreover, the operator S(t) defined in H0 maps H0 into itself and it enjoys the usual semigroup properties
(39)S(t+τ)=S(t)+S(τ),∀t,τ∈R,S(0)=I.
And it is obvious that the map {S(t),t≥0} is continuous in space H0.


## 4. The Existence of Global Attractor

In this section, we give the existence of a global attractor.A global attractor for aC0-semigroup S(t) defined on a complete metric space H is a bounded closed subset A⊂H which is positive fully invariant, that is, StA=A, for  all t≥0, and uniformly attracting, that is,
(40)dist⁡StB,A=sup⁡x∈S(t)Binf⁡y∈Adx,y⟶0ast⟶∞,
for any bounded set B⊂H.A bounded setB⊂H is an absorbing set for S(t) if, for any bounded set B⊂H, there exists tB=t(B)≥0 such that
(41)S(t)B⊂B,∀t≥tB,
which defines (H,S(t)) as a dissipative dynamical system.Theorem 4.
Assume the hypotheses of Theorem2 and ϱ=r=0, k>0 is sufficiently small, and then the corresponding semigroup S(t) of problem (1)–(4) has an absorbing set in H0.Proof.
Now, we show that semigroupS(t) has an absorbing set in H0. Firstly, we can calculate the total energy functional
(42)Et=12ut2+uxt2+uxx2+σ^ux2+f^uL,t-∫0L‍qutdx.
Let us fix an arbitrary bounded set B⊂H0 and consider the solutions of problem (1)–(4) given by (u(t),ut(t))=S(t)(u0,u1) with (u0,u1)∈B. Our analysis is based on the modified energy functional
(43)E~t=Et+L4q2+L0.
It is easy to see that (d/dt)E(t)=(d/dt)E~(t).
Indeed, sinceu(0)=0, we have
(44)uL,t2=uL,t-u0,t2=∫0L‍uxdx2≤∫0L‍dx∫0L‍ux2dx=Lux2.
In a similar way, since u(0,t)=ux(L,t)=uxx(0,t)=0, the following inequalities hold
(45)u≤Lux,ux≤Luxx,u≤L2uxx.
Now let us define
(46)φ(t)=∫0L‍ut-uxxtudx.
By multiplying (1) by ut and integrating over (0,L), we have
(47)ddtEt+ϕux2uxt2+gutl,tutl,t=0.
Then, multiplying (1) by u and integrating over (0,L), we obtain
(48)ddtφt-ut2-uxt2+uxx2+u(L)uxxx(L)+σux2ux2+ϕux2∫0L‍uxuxtdx=∫0L‍qudx.
Taking into account the boundary condition (3), we get
(49)ddtφ(t)-ut2-uxt2+uxx2+uLfuL,t+gutL,t+σux2ux2+ϕux2∫0L‍uxuxtdx=∫0L‍qudx.
Taking the sum of (47) with ε times (49) and using the Schwarz inequality, we have
(50)ddtEt+εddtφ(t)+εE(t)+ε2ut2+ε2uxt2+ε2uxx2+εσux2ux2-ε2σ^ux2+ϕux2uxt2+gutL,tut(L,t)=2εut2+uxt2-εϕux2∫0L‍uxuxtdx+εf^uL,t-εfuL,tu(L,t)-εgutL,tuL,t.
So, using (15), we obtain
(51)gutL,tutL,t≥k1utL,t2
and using (16) and (44), we obtain
(52)εgutL,tuL,t≤3εk2utL,tuL,t≤3L3/2εk2εuxxutL,t≤9εk22L3utL,t2+ε4uxx2.
Taking into account (18) and (44)-(45), we have
(53)εf^uL,t-εfuL,tuL,t≤ε2fuL,tuL,t+εL1≤ε23ku∞2+εL1≤ε23kL3uxx2+εL1.
Using the mean value inequality, we get
(54)-εϕux2∫0L‍uxuxtdx≤ε2ϕux2ux2+uxt2.
Then inserting (51)–(54) into (50), we obtain
(55)ddtE(t)+εddtφ(t)+εE(t)+ε2ut2+ε2uxt2+ε2uxx2+εσux2ux2-ε2σ^ux2+ϕux2uxt2+k1utL2≤2εut2+uxt2+ε2ϕux2uxt2+ε2ϕux2ux2+ε3kL3uxx2+εL1+9εk22L3utL,t2+ε4uxx2.
Taking 0<ε≤min⁡{k1/k22L3,1}=ε0 small enough, we get
(56)k1utL,t2>9εk22L3utL,t2.
Since with k>0 sufficiently small we have 1-6kL3≥0, we get
(57)ε2uxx2≥12ε3kL3uxx2+ε4uxx2.
Using ϕ(s)≥α+βsγ, (γ≥1, for all s∈R), we get
(58)12ϕux2uxt2≥α2uxt2+β2ux2γuxt2≥α2uxt2≥α4Lut2+α4uxt2.
So setting 0<ε≤min⁡{ε0,α/8L} small enough,
(59)α4Lut2+α4uxt2≥2εut2+uxt2.
Considering that σ(z)z-σ^(z)≥0 and σ(s)≥ϕ(s), for all s∈R, we obtain
(60)εσux2ux2-ε2σ^ux2-ε2ϕux2ux2≥0.
Therefore with (56)–(60), (55) is transformed into
(61)ddtEt+εddtφt+εEt≤εL1.
Adding εL0+εL4h2 on both sides of inequality (61) and taking into account that (d/dt)E(t)=(d/dt)E~(t), we obtain
(62)ddtE~t+εφt+εE~t≤εL0+L1+L4q2.
Now, let us set
(63)E~ε(t)=E~(t)+εφ(t).
Then since from (45)-∫0L‍qudx≥-L2quxx≥-L4q2-(1/4)uxx2, also f^(u(L,t))≥-L0 and σ^(ux2)≥0, we can get
(64)E~t≥14ut2+uxt2+uxx2=14ut,uttV×U2.
Thus, E~(t) dominates (u(t),ut(t))V×U. Also, from (45), we have
(65)E~εt-E~t=εφt=ε∫0L‍ut-uxxtudx≤εL2+L42ut2+uxt2+uxx2≤2εL2+L4E~(t)
which implies that, for ε sufficiently small enough,
(66)1-2εL2+L4E~t≤E~εt≤1+2εL2+L4E~t.
Inserting (66) into (62), we get
(67)ddtE~εt+11+2εL2+L4E~εt≤εL0+L1+L4q2.
Applying Gronwall’s inequality, we obtain
(68)E~εt≤E~ε0exp⁡-11+2εL2+L4t=ε∫0L‍ut-uxxtudx+εL0+L1+L4q21+2εL2+L4·1-exp⁡-11+2εL2+L4t.
Since the given invariant set B is bounded, E~ε(0) is also bounded. Then there exists tB>0 large enough such that
(69)1-2εL2+L4E~(t)≤E~ε(t)≤εL0+L1+L4q21+2εL2+L4,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀t>tB.
Then from (64) we have
(70)ut,uttV×U2≤4εL0+L1+L4q21+2εL2+L41-2εL2+L4,hhhhhhhhhhhhhhhhhhhhhhhhhhh∀t>tB.
This shows that
(71)B=u,ut∈H0:4εL0+L1+L4q21+2εL2+L41-2εL2+L4hhu,utH02hhh≤4εL0+L1+L4q21+2εL2+L41-2εL2+L4
is an absorbing set for S(t) in H0. The proof of Theorem 4 is ended.A semigroupS(t) is asymptotically smooth in H if for any bounded positive invariant set B⊂H, there exists a compact set K⊂B- such that
(72)dist⁡StB,K⟶0ast⟶∞.
Then the following lemma is well known.Lemma 5 (see [16], Theorem 2.3).
LetS(t) be a dissipative C0-semigroup defined on a metric space H; then S(t) has a compact global attractor in H if and only if it is asymptotically smooth in H.
The asymptotic smoothness can be verified from a result by Khanmamedov [17] and Chueshov and Lasiecka [16]. Assume that H is a Banach space.Lemma 6 (see [16], Proposition 2.10).
Assume that for any bounded positive invariant setB⊂H and for any ε>0, there exists T=T(ε,B) such that
(73)S(T)x-S(T)y≤ε+ϕTx,y,∀x,y∈B,
where ϕT:H×H→R satisfies
(74)liminf⁡k→∞liminf⁡l→∞ϕTzk,zl=0
for any sequence (zn) of B. Then S(t) is asymptotically smooth in H.Theorem 7.
Assume the hypotheses of Theorem2 and ϱ=r=0; then the corresponding semigroup S(t) of problem (1)–(4) is asymptotic compactness.Proof.
We are going to apply Lemma6 to prove the asymptotic compactness. Given initial data (u0,u1) and (v0,v1) in a bounded invariant set B⊂H0, let u, v be the corresponding weak solutions of problem (1)–(4). Then the difference w=u-v is a weak solution of
(75)wtt-wxxtt+wxxxx-σux2wxx-Δσvxx-ϕux2wxxt-Δϕvxxt=0,w(0,t)=wx(L·t)=wxx(0,t)=0,wxxxL,t=Δf+Δg,
where
(76)Δσ=σux2-σvx2,Δϕ=ϕux2-ϕvx2,Δf=fuL,t-fvL,t,Δg=gutL,t-gvtL,t.
Let us assume
(77)F(t)=12wt2+12wxt2+12wxx2+12σux2wx2
and define
(78)ψt=∫0L‍wt-wxxtwdx.
As before, by density, we can assume formally that w is sufficiently regular. Then, multiplying the first equation in (75) by wt and integrating over (0,L), we get
(79)12ddtwt2+wxt2+wxx2+σux2wx2+ϕux2wxt2+wxxxL,twtL,t=σ′ux2∫0L‍uxuxtdxwx2+Δσ∫0L‍vxxwtdx+Δϕ∫0L‍vxxtwtdx.
Taking into account the third equation in (75) we see that
(80)ddtF(t)+ϕux2wxt2+ΔgwtL,t=σ′ux2∫0L‍uxuxtdxwx2+Δσ∫0L‍vxxwtdx+Δϕ∫0L‍vxxtwtdx-ΔfwtL,t.
By multiplying first equation in (75) by w and integrating over (0,L), we obtain that
(81)ddtϕ(t)-wt2-wxt2+wxx2+σux2wx2+ϕux2∫0L‍wxwxtdx+wxxxwL,t-Δσ∫0L‍vxxwdx-Δϕ∫0L‍vxxtwdx=0.
Also taking into account the third equation in (75) we see that
(82)ddtϕ(t)+2F(t)=-ϕux2∫0L‍wxwxtdx-Δf+ΔgwL,t+Δσ∫0L‍vxxwdx-Δϕ∫0L‍vxtwxdx+2wt2+2wxt2.
Then summing (80) with η times (82) we obtain that
(83)ddtF(t)+ηddtϕ(t)+ηF(t)+ηF(t)+ϕux2wxt2+Δgwt(L,t)=σ′(ux2)∫0L‍uxuxtdxwx2+Δσ∫0L‍vxxwtdx+Δϕ∫0L‍vxxtwtdx-ΔfwtL,t-ηϕux2∫0L‍wxwxtdx-η(Δf+Δg)w(L,t)+ηΔσ∫0L‍vxxwdx-ηΔϕ∫0L‍vxtwxdx+2ηwt2+2ηwxt2.
In view of assumption (17) of the function g, we obtain that
(84)ΔgwtL,t≥k1wtL,t2.
Also since ϕ(s)≥α+βsγ(α,β>0,  γ≥1), we see that
(85)ϕux2wxt2≥α+βux2γwxt2.
In the following, let us estimate the right hand side of (83). We recall that u, v, and w satisfy the estimate ut2+uxt2+uxx2+σ^(ux2)≤M1, then denoting by C0 a generic positive constant which depends only on B we can simplify several notations.
Firstly, sinceσ(·), ϕ(·)∈C1(R), ϕ(0)=0, and ux2≤C0, we get σ′(ux2)≤C0, ϕ(ux2)≤C0; then we have
(86)σ′ux2∫0L‍uxuxtdxwx2≤C0wx2;-ηϕux2∫0L‍wxwxtdx≤C0wxwxt.
From the mean value theorem and noting that σ(a2)-σ(b2)≤σ′sup⁡a2,b2a+ba-b and ϕ(a2)-ϕ(b2)≤ϕ′sup⁡a2,b2a+ba-b, we have
(87)Δσ∫0L‍vxxwtdx≤C0wxwt,ηΔσ∫0L‍vxxwdx≤C0wx2,Δϕ∫0L‍vxxtwtdx≤C0wxwxt,-ηΔϕ∫0L‍vxtwxdx≤C0wx2.
From assumption (19) of the function f and ϱ=0 and inequalities (44), we get
(88)-ΔfwtL≤3kwt∞wtl,t≤C0wxwtL≤C0wx2+k2wtL2,-ηwLΔf≤η3kw∞wL≤η3kw∞Lwx≤C0wx2.
From assumption (18) of the function g and r=0 and inequalities (44), we get
(89)-ηwL,tΔg≤η3k2wL,twtL,t≤ηLwx3k2wtL,t≤9η2Lk222k1wx2+k12wtL,t2.
Substituting (84)–(89) into (83) and using Schwarz inequality, we obtain that
(90)ddtFt+ηddtϕt+2ηFt+α+βuxγwxt2≤5C0+9η2Lk222k1+2C02ηwx2+3ηL+3ηwxt2.
With η≤α/(3L+3) sufficiently small enough, we have
(91)ddtFt+ηddtϕt+2ηFt≤5C0+9η2Lk222k1+2C02ηwx2.
Defining Fη(t)=F(t)+ηϕ(t) and considering (44)-(45), we have
(92)Fηt-Ft=ηϕt=η∫0L‍wt-wxxtwdx≤ηwtw+ηwxtwx≤ηwtL2wxx+ηwxtLwxx≤ηL2wt2+wxt2+wxx2≤ηL2Ft,
which implies for η<min⁡{1/L2,α/(3L+3)} sufficiently small enough that
(93)1-ηL2Ft≤Fηt≤1+ηL2F(t).
Let C1=5C0+9η2Lk22/2k1+2C02/η; then inserting (93) into (91), we obtain that
(94)ddtFηt+2η1+ηL2Fηt≤C1wx2,t≥0.
From Gronwall’s lemma, we get
(95)Fηt≤Fη0e-2η/1+ηL2t+C1∫0t‍e-η/1+ηL2t-swxs2ds.
On the other hand, we have
(96)Fηt≥1-ηL2Ft=1-ηL22wt2+wxt2+wxx2hhhhhhhhhhh+σ(ux2)wx2≥1-ηL22wt2+wxt2+wxx2.
Therefore, combining (95) and (96), we can fix a constant CB>0, depending on the size of B but not on t>0, such that E1(0)≤CB; so
(97)wt,wttV×U≤21-ηL2CBexp⁡-η1+ηL2t+2C11-ηL2∫0t‍wxs2ds1/2.
Given ε>0, we choose T large such that
(98)21-ηL2CBexp⁡-η1+ηL2t≤ε
and define ϕT:H0×H0→R as
(99)ϕTu0,u1,v0,v1=2C11-ηL2∫0T‍uxt-vxt2dt1/2.
Then from (97)–(99), we get
(100)STu0,u1-STv0,v1H0≤ε+ϕTu0,u1,v0,v1
for all (u0,u1),(v0,v1)∈B.
Let(un0,un1) be a given sequence of initial data in B. Then the corresponding sequence (un(t),utn) of solutions of problem (1)–(4) is uniformly bounded in H0, because B is bounded and positively invariant. So {un} is bounded in C([0,∞),V)∩C1([0,∞),U). Since V↪H01(Ω) compactly, there exists a subsequence unk which converges strongly in C([0,T],H01(Ω)). Therefore
(101)lim⁡k→∞lim⁡l→∞∫0T‍uxnks-uxnls2ds=0,lim⁡k→∞lim⁡l→∞ϕTunk0,unk1,unl0,unl1=0.
So S(t) is asymptotically smooth in H0. That is, Lemma 6 holds. Thus Theorem 7 is proved.Remark 8.
The novelty and difficulty of Theorems4 lie in the appropriate definition on the functions φ(t)=∫0L‍ut-uxxtudx, ψ(t)=∫0L‍(wt-wxxt)wdx and the relationship between E~ε(t) and the energy function E~(t) and the relationship between Fη(t) and the function F(t).Lemma 9 ([18], Theorem 1.1 in Chapter I).
One assumes thatH is a metric space and that the operators {S(t)}t≥0 are given and satisfy the usual group properties and S(t) is continuous operator from H into itself and the semigroup {S(t)}t≥0 is asymptotically compact. One also assumes that there exist an open set U and a bounded set B of U such that U is absorbing in B. Then ω(B) is a compact attractor.By view of Lemma9, with Theorems 4 and 7, the main result of a global attractor reads as follows.Theorem 10.
The corresponding semigroupS(t) of problem (1)–(4) has a compact global attractor in the phase space H0.



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*Source: 857920-2015-02-22.xml*