# Asymptotic Smoothing and Global Attractors for a Class of Nonlinear Evolution Equations

Authors: Yongqin Xie; Zhufang He; Chen Xi; Zheng Jun
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations inH01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.


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## Body

## 1. Introduction

In this paper, we study the asymptotic regularity and the long-time behaviors of the solutions for the following semilinear evolution equations:(1)(|ut|r-2ut)t-Δu-μΔut-Δutt+f(u)=g,inΩ,u|t=0=u0,ut|t=0=u1inΩ,u=0on∂Ω,
where Ω is an open-bounded set of ℝ3 with smooth boundary ∂Ω, μ>0, and r satisfies
(2)3≤r≤6.Equation (1), which appears as a class of nonlinear evolution equations as r=2, is used to represent the propagation problems of lengthways wave in nonlinear elastic rods and ion-sonic of space transformation by weak nonlinear effect (see, for instance, [1–4]):
(3)utt-Δu-μΔut-Δutt+f(u)=g,
and as r=3 (named the Karman equation) is used to represent the flow of condensability airs in the across velocity of sound district (see, for instance, [5])
(4)ututt-uxx=0.
In [6], the authors have discussed the existence of global solutions in H01(Ω)×H01(Ω) under the assumptions that the initial values are sufficiently small. In [7, 8], the authors have discussed the nonexistence of global week solutions for the following system:
(5)P(ut)t+Au+Q(t,ut)=f(u),
where P, A, and Q are nonlinear operators. As r=2, in [9], the authors have discussed the long-time behaviors of solutions of (1) in H01(Ω)×H01(Ω); specifically in [10], the authors have discussed the long-time behaviors of solutions of (1) in W01,p(Ω)×W01,p(Ω). However, an open question remains whether the global attractor regularizes in the critical case and as 3≤r≤6, the long-time behaviors of solutions of (1) have not been considered completely up to now. In this paper, we try to discuss the problem.In the study of the global attractor regularization, the critical nonlinearity exponent brings a difficulty. About the regularity of attractor for the strongly damped wave equations, for the subcritical case, the authors in [11] have proved that the global attractor is bounded in H2×H1. For the critical case, Pata and Zelik [12] have proved that the global attractor is bounded in H2×H2 when the nonlinearity f(·) satisfies liminf|s|→∞(f′(s)/s)>-λ1, for all s∈ℝ, and the authors also pointed out further that one can prove the regularity of the attractor when f(·) only satisfies the natural assumptions (which have been realized recently in [13, 14]). A general way is to obtain higher regularity of the solutions than their initial values (see, for instance, [11, 12]). Then we can get the global attractor regularization. However, since (1) contains terms (|ut|r-2ut)t(r>2) and Δutt, they are essentially different from usual strongly damped wave equations and it is very difficult for us to obtain better regularity of the solutions of (1). Therefore, we must propose a new and more general way to study the smoothing of global attractors for (1).In this paper, we will apply the techniques introduced in Zelik [15], Sun and Yang [13], and Yang and Sun [14] to overcome the difficulty due to the critical nonlinearity and establish the asymptotic regularity of solutions. Based on this regularity result, we obtain asymptotically compactness of the semigroup {S(t)}t≥0 in H01(Ω)×H01(Ω), and the existence of the compact attractor 𝒜 has been proved. Moreover the compact attractor 𝒜 is bounded in H2(Ω)×H2(Ω).


## 2. Functional Setting

In what follows, we give some notations which will be used throughout this paper. LetΩ be a bounded subset of ℝ3 with a sufficiently smooth boundary ∂Ω, V=H01(Ω) and H=L2(Ω) corresponding norms, ∥u∥0=(∫Ω‍|∇u|2)1/2, and |u|2=(∫Ω‍|u|2)1/2, respectively; the norms in Lp(Ω)(3≤p<∞) are denoted by |u|p=(∫Ω‍|u|p)1/p. Let A be the (strictly) positive operator on L2(Ω) defined by
(6)A=-ΔwithdomainD(A)=H2(Ω)∩H01(Ω).For the family of Hilbert spacesD(As/2), s∈ℝ, their inner products and norms are, respectively,
(7)〈·,·〉s=〈As/2,As/2〉,∥·∥D(As/2)=|As/2|2.
Then we have the continuous embedding
(8)D(As/2)↪D(Ar/2)foranys>r,D(As/2)↪L6/(3-2s)(Ω)foranys∈[0,32).The product Hilbert spaces(9)Es=D(A(1+s)/2)×D(A(1+s)/2),
endowed with the usual inner products and norms
(10)∥(u,v)∥Es2=|A(1+s)/2u|22+|A(1+s)/2v|22,∀(u,v)∈Es.Denote byC any positive constant, which may be different from line to line and even in the same line.Throughout the paper we assume that the functionf∈ℂ1(ℝ,ℝ) satisfies the following conditions:
(11)|f′(s)|≤C(1+|s|4),∀s∈ℝ.
Also, let f admit the decomposition f=f0+f1, where f0,f1∈ℂ1(ℝ), satisfying
(12)|f0(s)|≤C(1+|s|5),∀s∈ℝ,(13)f0(s)s≥0,∀s∈ℝ,(14)|f1(s)|≤C(1+|s|p),p<5,∀s∈ℝ,(15)limsup|s|→∞f1(s)s>-λ1,
where λ1 is the first eigenvalue of -Δ in H01(Ω) with the Dirichlet boundary condition. Without loss of generality, we can think p is large enough, say, p≥3. Notice that by (13) and (15), there exists λ<λ1, such that
(16)f(s)s≥f1(s)s≥-λs2-C.We denote byF the function
(17)F(u)=∫Ω‍∫0u‍f(s)ds,
which is easily seen to satisfy the inequalities
(18)F(u)≥-12λ|u|22-C,(19)∫Ω‍f(u)u≥F(u)-12λ|u|22-C.We will complete our task exploiting the transitivity property of exponential attraction [16, Theorem 5.1], which we recall below for the readers’ convenience.Lemma 1 (see [16]).
Let𝒦1, 𝒦2, and 𝒦3 be subsets of E0 such that
(20)distE0(S(t)𝒦1,𝒦2)≤L1e-v1t,distE0(S(t)𝒦2,𝒦3)≤L2e-v2t,
for some v1,v2>0 and L1,L2≥0. Assume also that for all z1,z2∈⋃t≥0S(t)𝒦j(j=1,2,3) there holds
(21)∥S(t)z1-S(t)z2∥E0≤L0ev0t∥z1-z2∥E0,
for some v0≥0 and some L0≥0. Then it follows that
(22)distE0(S(t)𝒦1,𝒦3)≤Le-vt,
where v=v1v2/(v0+v1+v2) and L=L0L1+L2.


## 3. Global Solution inH01(Ω)×H01(Ω)

Exploiting the dissipative conditions (16) and (18) and the standard energy estimates technique, it is easy to obtain the following result.Lemma 2.
Assume that(23)∥(u0,u1)∥0<R
for some R>0. Then one has the following estimates: (i)
|ut(t)|rr+∥ut(t)∥02+∥u(t)∥02+∫0t‍∥ut(s)∥02ds≤Λ1(R);(ii)
∥utt(t)∥02+∫0t‍∥utt(s)∥02ds≤Λ2(R),for any t>0; here Λ1(R) and Λ2(R) are constants, which are dependent of R.Proof.
Multiplying (1) by ut, integrating in dx over Ω, and then integrating in dt on [0,t], we have
(24)|ut(t)|rr+∥ut(t)∥02+∥u(t)∥02+∫0t‍∥ut(s)∥02ds≤Λ1(R).
Multiplying (1) by utt and integrating by parts, we get
(25)∫Ω‍(r-1)|ut|r-2|utt|2+∥utt∥02≤∫Ω‍|∇u||∇utt|+μ∫Ω‍|∇ut||∇utt|+∫Ω‍|f(u)||utt|+∫Ω‍|g(x)||utt|.
By Holder’s and Young’s inequality associated with the Sobolev embedding theorem, we find that (25) implies that
(26)12∥utt∥02≤C∥u∥010+2μ2∥ut∥02+C|g(x)|22+Cmes(Ω),
where mes(Ω) denotes the measure of Ω.
Multiplying (1) by utt and integrating by parts, we get
(27)∥utt∥02+ddt[∫Ω‍∇u∇ut+μ∫Ω‍|∇ut|2+∫Ω‍f(u)ut+∫Ω‍g(x)ut]≤∫Ω‍|∇ut|2+∫Ω‍f′(u)|ut|2.
Combining with (11), we are led to the following estimation:
(28)∫Ω‍f′(u)|ut|2≤C∥u∥04∥ut∥02.
Applying (i), estimation (ii) follows by denotingΛ2=C(Λ1+|g(x)|22+mes(Ω)).
Based on the estimations above, the existence and uniqueness of the global weak solutions for (1) with the initial conditions can be obtained by standard Faedo-Galerkin method, which we omit here (see examples in Evans [17]).Theorem 3.
LetΩ⊂ℝ3 be a bounded domain. The assumptions (11)–(19) hold. u0(x),u1(x)∈H01(Ω). g(x)∈L2(Ω). Then (1) have such weak solutions u(x,t): for all  T>0(29)u,ut,utt,∈L∞(0,T;H01(Ω));
for all φ∈H01(Ω),
(30)〈(|ut|r-2ut)t-Δu-μΔut-Δutt+f(u)-g(x),φ〉=0,a.e.t∈[0,T].
Furthermore,
(31)u|t=0=u0(x),ut|t=0=u1(x),inH01(Ω).The weak solutions of (1) are unique and continuously dependent on initial conditions.Lemma 4.
LetΩ⊂ℝ3 be a bounded domain. The assumptions (11)–(19) hold. Let z0i(x)∈E0, i=1,2, be two initial values, g(x)∈L2(Ω), and denote by zi the two corresponding solutions to problem (1) in time interval I=[0,T]. Then the following estimate holds, for all t∈[0,T],
(32)∥z1(t)-z2(t)∥E02≤Q(∥z01∥E0,∥z02∥E0)eαt(∥z01-z02∥E02),
the constant α is only dependent on μ but independent of  T, t, and z0i(x)(i=1,2).From Theorem3 and Lemma 4, the initial boundary value problem (1) is equivalent to a continuous semigroup {S(t)}t≥0 defined by
(33)S(t):E0⟶E0,S(t):(u0,u1)↦(u,ut).


## 4. Bounded Absorbing Set

We now deal with the dissipative feature of the semigroup{S(t)}t≥0. Namely, we show that the trajectories originating from any given bounded set eventually fall, uniformly in time, into a bounded absorbing set B0⊂E0.Theorem 5.
{
S
(
t
)
}
t
≥
0 has a bounded absorbing set B0 in E0; that is, for any bounded subset B⊂E0, there exists T0=T0(B) such that
(34)S(t)B⊂B0∀t≥T0.Proof.
We setv=ut+δu and rewrite (1) as follows:
(35)(|ut|r-2ut)t+(μ-δ)Av+Avt+(1-μδ+δ2)Au=-f(u)+g(x),
and we take the L2 inner product of (35) with v and set δ small enough and
(36)E1(t)=r-1r|ut|rr+δ∫Ω‍|ut|r-2utu+12∥v∥02+12(1-μδ+δ2)∥u∥02+F(u),E2(t)=δ(1-μδ+δ2)∥u∥02+(μ-δ)∥v∥02+δ∫Ω‍f(u)u+δ|ut|rr-∫Ω‍g(x)v,
then we obtain
(37)E1(t)=-∫0t‍E2(s)ds+2δ∫0t‍|ut(s)|rrds+E1(0).
Young’s inequality gives that there exist positive constantsC1, C2, and k1 only dependent on Ω, δ such that
(38)E1(t)≥C1(r-1r|ut|rr+∥v∥02+ϖ∥u∥02)-C1ϖ|ut|r2(r-1)-k1;E2(t)≥C2(r-1r|ut|rr+∥v∥02+ϖ∥u∥02)-k2,
here ϖ=1-μδ+δ2-(λ/λ1)-(δϵ/λ1)>0 as δ small enough and k2=(|g|2).
By Lemma2, this shows that
(39)C1(r-1r|ut|rr+∥v∥02+ϖ∥u∥02)-k1≤-∫0t‍C2(r-1r|ut|rr+∥v∥02+ϖ∥u∥02-k2C2)+2δ∫0t‍|ut(s)|rrds+C11ϖ|ut|r2(r-1)+E1(0)≤-∫0t‍C2(r-1r|ut|rr+∥v∥02+ϖ∥u∥02-k2C2)+Λ(R)(r-2)/r(2δ∫0t‍|ut(s)|r2ds+C11ϖ|ut|rr)+E1(0).
Then we get(40)C1(r-1r|ut|rr+∥v∥02+ϖ∥u∥02)-k1≤-∫0t‍C2(r-1r|ut|rr+∥v∥02+ϖ∥u∥02-k2C2)+θΛ(R)+E1(0),
for any M>k2/C2 and θ=max{2δ,C1/ϖ}.
Hence, there existsT0=T0(B) such that
(41)r-1r|ut(T0)|rr+∥v(T0)∥02+ϖ∥u(T0)∥02≤M.
According to Lemma2, it follows that if t≥T0(B), then there exists M>0, such that
(42)|ut|rr+∥v∥02+∥u∥02≤M,
which completes the proof of the desired results.Corollary 6.
For anyB⊂E0, there exists T=T(B), such that
(43)∫st‍(∥ut(τ)∥02+∥utt(τ)∥02)dτ+∥ut(t)∥02+∥utt(t)∥02≤M1,
provided that t≥s≥T, where M1 is a positive constant only dependent on C (given by (16) and (18)) and M (given by (42)).Proof.
Multiplying (1) by ut(t),utt, respectively, integrating in dx over Ω, and then integrating in dt on [s,t] and associated with Theorem 5 and Sobolev embedding H01↪Lr, we can get the conclusions above.Hereafter, we always assume that for someδ>0andv=ut+δu,
(44)B0={(u,v)∈E0;∥u∥+∥v∥≤M}
is the bounded absorbing set of {S(t)}t≥0 in E0 obtained in Theorem 5.


## 5. Asymptotic Regularity of the Solutions

In this section, we will establish some a priori estimates about the solutions of (1), which are the basis of our analysis. Let u(x,t) be a unique weak solution of (1) corresponding to the initial data z0=(u0,u1)∈H01(Ω)×H01(Ω). We decompose u into the sum
(45)u(t)=v(t)+w(t),
where v(t) and w(t) are the solutions to the problems
(46)(|vt|r-2vt)t-Δv-μΔvt-Δvtt+f0(v)=0,v|∂Ω=0,v(0)=u0,vt(0)=u1,(47)(|ut|r-2ut-|vt|r-2vt)t-Δw-μΔwt-Δwtt+f(u)-f0(v)=g,w|∂Ω=0,w(0)=0,wt(0)=0.It is convenient to denote(48)z(t)=(u(t),ut(t)),zd(t)=(v(t),vt(t)),zc(t)=(w(t),wt(t)).Lemma 7.
For anyR≥0, there exist M2=M2(R)≥0 and k0>0, such that whenever ∥z0∥0≤R, it follows that
(49)∥zd(t)∥0≤M2e-k0t,∀t∈ℝ+,
the constant k0 is independent of R and t.Proof.
Denotingξ=vt+δv and repeating word by word Theorem 5, that applies to the present case with zd(t) in place of z(t) (with the further simplification that C=0 for now f1=0,g=0).
Set(50)E(t)=12∥ξ∥02+12(1-μδ+δ2)∥v∥02+F0(v).
Let δ≤min(μ/2,1/μ) be small enough and Cδ=(1/2)max(1,1-μδ+δ2)>0; then
(51)E(t)≥Cδ(∥v∥02+∥ξ∥02).
Combining with (13) we get the differential inequality
(52)ddtE(t)+δ(1-μδ+δ2)∥v∥02+(μ-δ)∥ξ∥02≤(r-1)κr2|vt|r(r-2)∥vtt∥0∥ξ∥0-δ∫Ω‍f0(v)v≤(r-1)2κr42(μ-δ)|vt|r2(r-2)∥vtt∥02+μ-δ2∥ξ∥02≤Cμ|vt|r2(r-3)∥vtt∥02[∥ξ∥02+∥v∥02]+μ-δ2∥ξ∥02,
where Cμ=(r-1)2κr5/2(μ-δ) and κr is the Sobolev embedded constant H01↪Lr.
Multiplying (46) by vt(t), integrating in dx over Ω, and then integrating in dτ on [0,t],
(53)sup∥z0∥0≤Rsupt∈ℝ+∥zd(t)∥0<+∞.
So we get
(54)∫Ω‍F0(v)≤C(|v|2+|v|66)≤k∥v∥02,∀t≥0,
where k=k(R)≥1.
By (50),
(55)E(t)≤12∥ξ∥02+12(1+k-μδ+δ2)∥v∥02≤12(1+k-μδ+δ2)(∥ξ∥02+∥v∥02),
and by the above
(56)δ(1-μδ+δ2)∥v∥02+12(μ-δ)∥ξ∥02≥α(∥ξ∥02+∥v∥02),
here α=min{δ(1-μδ+δ2),(1/2)(μ-δ)}. Then there exists a constant k0(=2α/(1+k-μδ+δ2)) such that
(57)k0E(t)≤δ(1-μδ+δ2)∥v∥02+12(μ-δ)∥ξ∥02.
Combining (51), (57), and Young’s inequality, we are led to the differential inequality
(58)ddtE(t)≤(-k0+CμCδ|vt|r2(r-3)∥vtt∥02)E(t).
By Gronwall Lemma, we get
(59)E(t)≤E(0)e-k0te∫0t(Cμ/Cδ)‍|vt(s)|r2(r-3)∥vtt(s)∥02ds∀t≥0.
Due to Lemma2 and (46), then ∥zd(0)∥0=∥z0∥<R; for any t>0, there holds
(60)E(t)≤E(0)e(Cμ/Cδ)Λ12(r-3)/rΛ2e-k0t∀t≥0.
From (57) we get
(61)E(0)≤ϖω∥z0∥02,
where ϖ=min{δ(1-μδ+δ2),(1/2)(μ-δ)}. Putting together (51), the proof is finished.Lemma 8.
For anyR≥0, there exist M3=M3(R)≥0 and k1=k1(R)>0, such that whenever ∥z0∥0≤R and for any time t∈ℝ+ and every μ, the solutions of (47) satisfy the following estimates:
(62)∥zc(t)∥σ2=|A(1+σ)/2w(t)|22+|A(1+σ)/2wt(t)|22≤M3(1+|g|22)ek1t.Proof.
Due to Lemma2, Theorem 5, and Corollary 6, there exists an increasing function Q1(·) in [0,+∞), such that
(63)|ut|rr+|vt|rr+∥utt∥02+∥vtt∥02+∥u∥02+∥v∥02≤Q1(R),∀t∈ℝ+.
Choosing(64)σ=min{14,5-p2}
and multiplying (47) by Aσwt, we are led to the identity
(65)12ddt∥zc∥σ2+μ∥A(1+σ)/2wt∥2=-〈(|ut|r-2ut-|vt|r-2vt)t,Aσwt〉-〈f(u)-f(v),Aσwt〉-〈f1(v),Aσwt〉+〈g,Aσwt〉,
where
(66)∥zc∥σ2=|A(1+σ)/2w|22+|A(1+σ)/2wt|22.
Combining with(r-2)/(4-2σ)≤1 and (2), we have
(67)〈(|ut|r-2ut-|vt|r-2vt)t,Aσwt〉≤(r-1)∫Ω‍(|ut|r-2|utt|+|vt|r-2|vtt|)|Aσwt|≤C(|ut|6(r-2)/(4-2σ)r-2|utt|6+|vt|6(r-2)/(4-2σ)r-2|vtt|6)|Aσwt|6/(1+2σ)≤C(∥ut∥0(r-2)∥utt∥0+∥vt∥0(r-2)∥vtt∥0)|A(1+σ)/2wt|2≤CQ1(R)2(r-1)+μ4|A(1+σ)/2wt|22.
By virtue of (11), we get
(68)-〈f(u)-f(v),Aσwt〉≤C∫Ω‍(1+|u|4+|v|4)|w||Aσwt|≤C(1+|u|64+|v|64)|w|6/(1-2σ)|Aσwt|6/(1+2σ)≤C(1+∥u∥04+∥v∥04)∥A(1+σ)/2w∥∥A(1+σ)/2wt∥≤C(1+Q1(R)4)∥zc∥σ2+μ4∥A(1+σ)/2wt∥2.
Sincep/(5-2σ)≤1, by (14) we deduce that
(69)〈f1(v),Aσwt〉≤C∫Ω‍(1+|v|p)|Aσwt|≤C(1+|v|6p/(5-2σ)p)|Aσwt|6/(1+2σ)≤C(1+∥v∥0p)∥A(1+σ)/2wt∥≤C(1+Q1(R)p)+μ4∥A(1+σ)/2wt∥2.
Finally,(70)〈g,Aσwt〉≤C∥A-1/2g∥∥A(1+σ)/2wt∥≤C|g|22+μ4∥A(1+σ)/2wt∥2.
Plugging (67)–(70)  into  (65), we obtain
(71)ddt∥zc∥σ2≤C(1+Q1(R)4)∥zc∥σ2+C|g|22+Q2(R),
where Q2(·) is an increasing function in [0,+∞) and the Gronwall lemma entails, there exists a constant M3=M3(R) such that
(72)∥zc∥σ2≤M3(R)(1+|g|22)ek1t∀t∈[0,T].
The proof is finished.Lemma 9.
LetΩ⊂ℝ3 be a bounded domain. The assumptions (11)–(15) hold, and let (u(t),ut(t)) be the solution of (1) corresponding to the initial data z0=(u0,u1). Then, for any γ>0, one can decompose u(t) as
(73)u(t)=v1(t)+w1(t)∀t≥0,
where v1(t) and w1(t) satisfy the following estimates: for all t≥0,
(74)∫st‍∥v1(τ)∥02dτ≤γ(t-s)+Cγ,(75)|A(1+σ)/2w1(t)|22≤Kγ,
with the constants Cε and Kγ depending on γ, ∥z0∥0, and |g|22 but both being independent of t.Proof.
From Theorem5, we know that there exists a constant M∥z0∥0 which depends only on the E0-bound of z0, such that
(76)supt≥0∥S(t)z0∥02≤M∥z0∥0.
Now, taking T≥(1/k0)ln(βE(0)/ε) and in every interval [mT,(m+1)T),m=1,2,…, we set
(77)v1(t)=v(t),w1(t)=w(t),
where v(t) and w(t) are the solutions of (46) and (47), respectively, in the interval [(m-1)T,(m+1)T) with the initial data (v((m-1)T),vt((m-1)T))=S(t)((m-1)T)z0 and (w((m-1)T),wt((m-1)T))=(0,0,0). And in interval [0,T), we set v1(t)=v(t) and w1(t)=w(t), where v(t) and w(t) are the solutions of (46) and (47), respectively, in the interval [0,T) with the initial data (v(0),vt(0))=z0 and (w(0),wt(0))=(0,0).
Then from Lemma7, we have
(78)∫st‍∥v1(τ)∥02dτ≤γ(t-s)+χ[0,T)M2∀0≤s≤t,
and from Lemma 8, we have
(79)|A(1+σ)/2w1(t)|22≤M3(R)(1+|g|22)e2k1T∀t≥0,
where χ[0,T) is the characteristic function of set [0,T).Remark 10.
From the proof of Lemma9, we observe that the decomposition v1(t) and w1(t) also satisfy further that
(80)∥v1(t)∥02≤M0(R)∀t≥0.
In what follows we begin to establish the asymptotic regularity of the solutions. Now we can claim the following result.Lemma 11.
LetΩ⊂ℝ3 be a bounded domain with smooth boundary, and assume that f satisfies (11)–(15); there exists constant 𝒥0 which depends only on the E0-bounds of B(⊂E0), such that for any t≥0 and z0∈B(81)∥zc(t)∥σ2≤𝒥0
holds.Proof.
Letϵ=min(1/4,μ/2) and multiplying (47) by Aσ(wt(t)+ϵw(t)), we get that
(82)12ddt[(1+μϵ)|A(1+σ)/2w|22+|A(1+σ)/2wt|22+2〈-Δwt,ϵAσw〉]+(μ-ϵ)|A(1+σ)/2wt|22+ϵ|A(1+σ)/2w|22=-〈(|ut|r-2ut-|vt|r-2vt)t,Aσ(wt+ϵw)〉-〈f(u)-f(v),Aσ(wt+ϵw)〉-〈f1(v),Aσ(wt+ϵw)〉+〈g,Aσ(wt+ϵw)〉.
Let(83)H(t)=(1+μϵ)|A(1+σ)/2w|22+|A(1+σ)/2w|22+2〈-Δwt,ϵAσw〉,
then
(84)H(t)≤ϖ1(|A(1+σ)/2w|22+|A(1+σ)/2w|22),
here ϖ1=max(1+(1+μ)ϵ,1+ϵ)>1 and
(85)H(t)≥ϖ2(|A(1+σ)/2w|22+|A(1+σ)/2w|22),
where ϖ2=min(1+(μ-1)ϵ,1-ϵ).
Combining with (2), (63), and (r-2)/(4-2σ)≤1, we have
(86)〈(|ut|r-2ut-|vt|r-2vt)t,Aσ(wt+ϵw)〉≤(r-1)∫Ω‍(|ut|r-2|utt|+|vt|r-2|vtt|)(|Aσwt|+ϵ|Aσw|)≤C(|ut|6(r-2)/(4-2σ)r-2|utt|6+|vt|6(r-2)/(4-2σ)r-2|vtt|6)×(|Aσwt|6/(1+2σ)+ϵ|Aσw|6/(1+2σ))≤C(∥ut∥0(r-2)∥utt∥0+∥vt∥0(r-2)∥vtt∥0)×(|A(1+σ)/2wt|2+ϵ|A(1+σ)/2w|2)≤CQ1(R)2(r-1)+μ-ϵ8|A(1+σ)/2wt|22+ϵ8|A(1+σ)/2w|22.
Using Lemma 8 to deal with the nonlinear term, we get
(87)-〈f(u)-f(v),Aσwt〉≤C∫Ω‍(1+|u|4+|v|4)|w||Aσwt|,∫Ω‍|u|4|w||Aσwt|≤∫Ω‍(|v1|4+|w1|4)|w||Aσwt|.
Using Remark10, we have
(88)∫Ω‍|v1|4|w||Aσwt|≤C(|v1|64)|w|6/(1-2σ)|Aσwt|6/(1+2σ)≤C∥v1∥04|A(1+σ)/2w|2|A(1+σ)/2wt|2≤CM0(R)2∥v1∥02×(|A(1+σ)/2w|22+|A(1+σ)/2wt|22).
Moreover, from Lemma2, Remark 10, and Lemma 8, there exists a constant M4=M4(R) such that, for all t≥0,
(89)∥zc(t)∥02=∥w(t)∥02+∥wt(t)∥02≤M4,∫Ω‍|w1|4|w||Aσwt|≤C|w1|6/(1-2σ)4|w|6/(1+6σ)|Aσwt|6/(1+2σ)≤C∥w∥02|A(1+σ)/2w1|24|A(1+σ)/2wt|2≤CM4Kγ2|A(1+σ)/2wt|2≤CM42Kγ4+μ-ϵ8|A(1+σ)/2wt|22,
where Kγ is given in (75).
At the same time, we get(90)∫Ω‍|v|4|w||Aσwt|≤CM∥v∥04(|A(1+σ)/2w|22+|A(1+σ)/2wt|22).
Similarly(91)-〈f(u)-f(v),Aσw〉≤CM0(R)2∥v1∥02|A(1+σ)/2w|22+C(1+M42Kγ4)+ϵ8|A(1+σ)/2w|22+CM∥v∥04|A(1+σ)/2w|22.
Since p/(5-2σ)≤1, by (14) and Lemma 7, we deduce that
(92)〈f1(v),Aσ(wt+ϵw)〉≤C∫Ω‍(1+|v|p)(|Aσwt|+ϵ|w|)≤C(1+|v|6p/(5-2σ)p)(|Aσwt|6/(1+2σ)+ϵ|Aσwt|6/(1+2σ))≤C(1+∥v∥0p)(|A(1+σ)/2wt|2+ϵ|A(1+σ)/2w|2)≤C(1+M2p)+μ-ϵ8|A(1+σ)/2wt|22+ϵ8|A(1+σ)/2w|22.
Finally,(93)〈g,Aσ(wt+εw)〉≤C|A-1/2g|2|A(1+σ)/2(wt+εw)|2≤C|g|22+μ-ϵ8|A(1+σ)/2wt|22+ϵ8|A(1+σ)/2w|22.
Moreover, following Lemma 7, we can take T1 large enough such that for all t≥T1(94)∥v(t)∥04≤min(ϵ16CM,μ-ϵ16CM).
Plugging (86)–(93) into (65), we obtain
(95)ddtH(t)+ϑH(t)≤CR∥v1∥2H(t)+C|g|22+Q2(R),
where
(96)ϑ=min(μ-ϵ,ϵ),CR=2CM0ϖ2,
and Q2(·)=C(1+Q1p+M42Kγ4) is an increasing function in [0,+∞).
Then using the Gronwall inequality and integrating over[T1,t], we entail
(97)H(t)≤e-∫T1t‍(ϑ-CR∥v1(s)∥02)dsH(T1)+(C|g|22+Q2(R))∫T1t‍e∫ts‍(ϑ-CR∥v1(τ)∥02)dτds.
Letting γ<ϑ/2CR, then we get
(98)H(t)≤e-(ϑ/2)(t-T1)H(T1)+2ϑ(C|g|22+Q2(R)).
Hence, by Lemma 8, (85) and noting that T1 is fixed, the proof is finished.Lemma 12.
Assume thatBσ is bounded in Eσ. Then there exists a constant Mσ(>0) which depends only on the Eσ-bounds of Bσ such that, for any t≥0 and z0∈Hσ(99)∥z(t)∥σ≤Mσ.Proof.
We setv=ut+δu, take the L2 inner product of (35) with Aσv, and set δ≤min(μ/2,1/μ); we have
(100)12ddt[|A(1+σ)/2v|22+(1-μδ+δ2)|A(1+σ)/2u|22]+(μ-δ)|A(1+σ)/2v|22+δ(1-μδ+δ2)|A(1+σ)/2u|22=-(r-1)〈|ut|r-2utt,Aσv〉-〈f(u),Aσv〉+〈g,Aσv〉.
We only need to deal with the nonlinear term separately, and the remainders are same as in Lemma 11:
(101)-(r-1)〈|ut|r-2utt,Aσv〉≤(r-1)|ut|6(r-2)/(4-2σ)r-2|utt|6|Aσv|6/(1+2σ)≤C∥ut∥02(r-2)∥utt∥02+μ-δ8|A(1+σ)/2v|2;〈g,Aσv〉≤C|g(x)|22+μ-δ8|A(1+σ)/2v|2;-〈f(u),Aσv〉≤C∫Ω‍(1+|u|5)|Aσv|≤C(1+∥v∥02)+C∫Ω‍|u|4|u||Aσv|,
Using Lemma9 and Remark 10, we have
(102)C∫Ω‍|u|4|u||Aσv|≤C∫Ω‍(|v1|4+|w1|4)|u||Aσv|,C∫Ω‍|v1|4|u||Aσv|≤C|v1|64|u|6/(1-2σ)|Aσv|6/(1+2σ)≤C∥v1∥04|A(1+σ)/2u|2|A(1+σ)/2v|2≤C∥v1∥02[|A(1+σ)/2u|22+(1-μδ+δ2)|A(1+σ)/2v|22],C∫Ω‍|w1|4|u||Aσv|≤C|u|6|w1|6/(1-2σ)4|Aσv|6/(1+2σ)≤C∥u∥0|A(1+σ)/2w1|24|A(1+σ)/2v|2≤C|A(1+σ)/2v|2≤C+μ-δ4|A(1+σ)/2v|22.
Substituting (101)–(102) into (100), we get that, for all t≥0,
(103)ddt[|A(1+σ)/2v|22+(1-μδ+δ2)|A(1+σ)/2u|22]+ω[|A(1+σ)/2v|22+(1-μδ+δ2)|A(1+σ)/2u|22]≤C∥v1(t)∥02[|A(1+σ)/2u|22+(1-μδ+δ2)|A(1+σ)/2v|22]+C(1+∥ut(t)∥02(r-2)∥utt(t)∥02+C|g(x)|22),
where ω=min((μ-δ)/2,2δ). Using the Gronwall inequality and integrating over [0,t], we get that there exists a constant Mσ(>0) which depends only on the Eσ-bounds of Bσ such that, for any t≥0 and z0∈Bσ,
(104)∥z(t)∥σ≤Mσ.In the following, based on Lemmas11 and 12, we perform a bootstrap argument, whose proof is similar to that of Lemmas 11 and 12 (e.g., see [11, 13]). Here we only point out the results and omit the proof.Lemma 13.
For eachσ≤θ≤1, assume that Bθ is bounded in Hθ. Then there exists a constant Mθ which depends only on the Hθ-bounds of Bθ, such that
(105)∥S(t)z0∥θ≤Mθ
holds for all t≥0 and z0∈Bθ.Lemma 14.
For eachσ≤θ≤1-σ, if the initial data set Bθ is bounded in Hθ, then the decomposed ingredient zc=w(t),wt(t) (the solutions of (47)) satisfies
(106)∥zc(t)∥θ+σ2≤𝒥θ
which holds for all t≥0 and z0∈Bθ, where the constant Jθ depends only on the Hθ-bounds of Bθ.Theorem 15 (asymptotic regularity of solutions).
LetΩ be a bounded domain in ℝ3 with smooth boundary and let f satisfy (11)–(15). Assume further that g(x)∈L2(Ω). Then there exist a bounded (in E1) set B1⊂E1, a positive constant ν, and a monotonically increasing function Q(·) such that, for any bounded (in E0) set B⊂E0 and t≥0, the following estimate holds:
(107)distE0(S(t)B,B1)≤Q(∥B∥0)e-νt,
where distE0 denotes the usual Hausdorff semidistance in E0.Proof.
LetB0 be a bounded (in E0) absorbing set obtained in Theorem 5.
From Lemmas11 and 7, we know that the set Aσ={z∈Eσ:∥z∥σ≤𝒥0} satisfies
(108)distE0(S(t)B0,Aσ)≤Q1(∥B0∥E0)e-k0t,
where the constant 𝒥0 is given in Lemma 11 corresponding to B0. Applying Lemmas 14 and 7 to Aσ, we see that there is a set A2σ which is bounded in E2σ, such that
(109)distE0(S(t)Aσ,A2σ)≤Q1(∥Aσ∥E0)e-k0′t,
where k0′ depends on the E0-bounds of Aσ (determined in Lemma 7).
Combining with Lemma4, we know that the conditions in Lemma 1 are all satisfied. Hence, we have
(110)distE0(S(t)B0,A2σ)≤Q1(∥B0∥E0)e-k0′′t,
for two appropriate constants C and k0′′. Since σ=min{1/4,(5-p)/2,(6-r)/2} is fixed, by finite steps (e.g., at most by [1/σ]+1 steps), we can infer that there is a bounded (not only in E0, but also bounded in E1) set B1⊂E1 such that
(111)distE0(S(t)B0,B1)≤Q(∥B0∥E0)e-νt.
Note further that all the constants in (111) depend only on ∥B0∥0 and |g|2. Now, for any bounded (in E0) B, from Theorem 5 we see that there is a TB such that
(112)S(t)B⊂B0,∀t≥TB.
Hence,
(113)distE0(S(t)B,B0)≤Me-ν(t-TB),
where M=sup{∥S(t)B∥0:0≤t≤TB}<∞. Finally, we apply the attraction transitivity lemma, that is, Lemma 1, again to (111) and (113), and this completes the proof.


## 6. Global Attractor

Collecting now Theorem5, Lemmas 7 and 8, and Theorem 15, we establish that {S(t)}t≥0 is asymptotically compact. Therefore, by means of well-known results of the theory of dynamical systems we get the following.Theorem 16.
LetΩ⊂ℝ3 be a bounded domain with smooth boundary and assume that f satisfies (11)–(15); the semigroup {S(t)}t≥0 possesses a global attractor 𝒜 on H01(Ω)×H01(Ω). Moreover, this attractor 𝒜 is bounded in H2(Ω)×H2(Ω).



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*Source: 989475-2013-05-21.xml*