# Strong Global Attractors for 3D Wave Equations with Weakly Damping

Authors: Fengjuan Meng
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

We consider the existence of the global attractor𝒜1 for the 3D weakly damped wave equation. We prove that 𝒜1 is compact in (H2(Ω)∩H01(Ω))×H01(Ω) and attracts all bounded subsets of (H2(Ω)∩H01(Ω))×H01(Ω) with respect to the norm of (H2(Ω)∩H01(Ω))×H01(Ω). Furthermore, this attractor coincides with the global attractor in the weak energy space H01(Ω)×L2(Ω).


---

## Body

## 1. Introduction

LetΩ⊂R3 be a bounded domain with smooth boundary ∂Ω. We consider the following weakly damped wave equation: (1.1)utt+αut-Δu+φ(u)=finΩ×R+
with the boundary condition(1.2)u∣∂Ω=0,
and initial conditions:(1.3)u(⋅,0)=u0,ut(⋅,0)=u1,inΩ,
where α>0, φ is the nonlinear term, and f is a given external forcing term.Nonlinear wave equation of the type (1.1) arises as an evolutionary mathematical model in many branched of physics, for example, (i) modeling a continuous Josephson junction with φ(u)=βsinu; (ii) modeling a relativistic quantum mechanics with φ(u)=|u|γu. A relevant problem is to investigate the asymptotic dynamical behavior of these mathematical models. The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem is to analyse the existence of its global attractor.The existence of global attractors for the classical wave equations inH01(Ω)×L2(Ω) and the regularities of the global attractors has been studied extensively in many monographs and lectures, for example, see [1–7] and references therein.However, to our knowledge, the research about the stronger attraction of global attractors for the damped wave equations with respect to the norm of(H2(Ω)∩H01(Ω))×H01(Ω) is fewer, only has been found in [8–10]. In the above three papers, the global attractors in strong topological space (H2(Ω)∩H01(Ω))×H01(Ω) were established, the attraction with respect to the norm of (H2(Ω)∩H01(Ω))×H01(Ω) was proved by the asymptotic compactness of the operator semigroup.Recently, we consider (1.1) in n dimensional space where the nonlinear term φ without polynomial growth is in [11].In this paper, our aim is to prove the existence of a global attractor for (1.1) in strong topological space (H2(Ω)∩H01(Ω))×H01(Ω) where the nonlinear term φ with some polynomial growth. For simplicity, we consider the space dimension is 3, as we know, when the space dimension is lagerer than 3, the case is similar as in 3D, when the space dimension is 1 or 2, the case is more easier. The attraction with respect to the norm of (H2(Ω)∩H01(Ω))×H01(Ω) will be proved by a method different from [8–10]. Furthermore, this attractor coincides with the global attractor in the weak energy space H01(Ω)×L2(Ω).The basic assumptions about the external forcing termf and the nonlinear term φ are as follows. Let f∈L2(Ω) be independent of time, and let the nonlinear term φ∈C1(R,R) satisfy the following assumptions:(1.4)lim⁡inf⁡|s|→∞Φ(s)s2≥0,hereΦ(s)=∫0sφ(τ)dτ;(1.5)limsup⁡|s|→∞|φ′(s)|s2=0;
moreover, there exists a constant C0>0 such that(1.6)lim⁡inf⁡|s|→∞sφ(s)-C0Φ(s)s2≥0.Throughout this paper, we use the following notations. LetΩ be a bounded subset of Rn with sufficiently smooth boundary, A=-Δ. V=H01(Ω), H=L2(Ω), and D(A)=H2(Ω)∩H01(Ω) with the corresponding norms ∥u∥=(∫Ω|∇u(x)|2dx)1/2, |u|=(∫Ω|u(x)|2dx)1/2 and |Δu|=(∫Ω|Δu(x)|2dx)1/2, respectively. The norms in Lp(Ω),1≤p<∞ are denoted by |u|p=(∫Ω|u|pdx)1/p, the scalar products of V,H are denoted by(1.7)((u,v))=∫Ω∇u(x)∇v(x)dx,(u,v)=∫Ωu(x)v(x)dx,
respectively. We have D(A)⊂V⊂H=H*⊂V*,H* and V* are the dual spaces of H and V, respectively, and each space is dense in the following one and the injections are continuous. Then, we introduce the product Hilbert spaces ℋ0=V×H=H01(Ω)×L2(Ω),ℋ1=D(A)×V=(H2(Ω)∩H01(Ω))×H01(Ω), endowed with the standard product norms:(1.8)‖{u1,u2}‖H02=‖u1‖V2+‖u2‖H2,‖{u1,u2}‖H12=‖u1‖D(A)2+‖u2‖V2.
Denote by C any positive constant which may be different from line to line and even in the same line, we also denote the different positive constants by Ci,i∈ℕ, for special differentiation.The rest of the paper is organized as follows. In the next section, for the convenience of the reader, we recall some basic concepts about the global attractors and recapitulate some abstract results. In Section3, we present our main results.


## 2. Preliminaries

In this section, we first recall some basic concepts and theorems, which are important for getting our main results. We refer to [2, 5, 6, 12, 13] and the references therein for more details. Then, we outline some known results about (1.1)–(1.3).Definition 2.1.
The mappingsS(t), where S:X×[0,+∞)→X, is said to be a C0 semigroup on X, if {S(t)}t≥0 satisfies(1)
S(0)u=u for all u∈X;(2)
S(t1)(S(t2)u)=S(t1+S2)(u) for all u∈X and t1,t2∈R+;(3)
the mappingS:X×(0,∞)→X is continuous.Definition 2.2.
Let{S(t)}t≥0 be a semigroup on a metric space (E,d). A subset 𝒜 of E is called a global attractor for the semigroup, if 𝒜 is compact and enjoys the following properties:(1)
𝒜 is invariant, that is, S(t)𝒜=𝒜, for  allt≥0;(2)
𝒜 attracts all bounded sets of E. That is, for any bounded subset B of E,(2.1)d(S(t)B,A)⟶0,ast⟶0,
where d(B,A) is the semidistance of two sets B and A:
(2.2)d(B,A)=sup⁡x∈Binf⁡y∈Ad(x,y).Definition 2.3.
AC0 semigroup {S(t)}t≥0 in a Banach space X is said to satisfy the condition (C) if for any ε>0 and for any bounded set B of X, there exist t(B)>0 and a finite dimensional subspace X1 of X such that {∥PS(t)x∥X,x∈B,t≥t(B)} is bounded and
(2.3)‖(I-P)S(t)x‖X<ε,t≥t(B),x∈B,
where P:X→X1 is a bounded projector.Definition 2.4.
Let{S(t)}t≥0 be a semigroup on a metric space (E,d). A set B0⊂E is called an absorbing set for the semigroup {S(t)}t≥0, if and only if for every bounded set B⊂E, there exists a T0=T0(B)>0 such that S(t)B⊂B0 for all t≥T0.Theorem 2.5.
LetX be a Banach space and let {S(t)}t≥0 be a C0 semigroup in X. Then, there is a global attractor for {S(t)}t≥0 in X if the following conditions hold true:(1)
{S(t)}t≥0 satisfies the condition (C), and(2)
there is a bounded absorbing setB⊂X.In [12], the authors have discussed the relations between Condition (C) and ω-limit compact and proved that, in uniformly convex Banach space, Condition (C) is equivalent to ω-limit compact, if the semigroup has a bounded absorbing set.Next, we recall the result about the global attractor inℋ0 whose proofs are omitted here, the reader is referred to [6] and the reference therein.Theorem 2.6.
Under the conditions (1.4), (1.5), (1.6), the solution semigroup {S(t)}t≥0 of the problem (1.1)–(1.3) has a global attractor 𝒜0 in ℋ0. 𝒜0 is included and bounded in ℋ1.


## 3. Main Results

According to the standard Fatou-Galerkin method, it is easy to obtain the existence and uniqueness of solutions and the continuous dependence to the initial value of (1.1)–(1.3). We address the reader to [6] and the reference therein. Here, we only state the result as follows.Lemma 3.1.
Let conditions (1.4), (1.5), (1.6) hold, then for any T>0 and (u0,u1)∈ℋ0, there exists a unique solution of (1.1)–(1.3) such that
(3.1){u,ut}∈C([0,T];H0).
If, furthermore,
(3.2)(u0,u1)∈H1,
then u satisfies
(3.3){u,ut}∈C([0,T];H1).We define the mappings:(3.4)S(t):{u0,u1}⟶{u(t),ut(t)}∀t∈R.
By Lemma 3.1, it is easy to see that {S(t)}t≥0 is C0 semigroup in the energy phase spaces ℋ0 and ℋ1.In order to verify the existence of the bounded absorbing set inℋ1, we need the result about the existence of the bounded absorbing set in ℋ0. First, we establish the bounded absorbing set in ℋ0. Its proof is essentially established in [6] and the reference therein, and we only need to make a few minor changes for our problem. Here, we only give the following lemma.Lemma 3.2.
Under the conditions (1.4), (1.5), (1.6), {S(t)}t≥0 has a bounded absorbing set ℬ0≜Bℋ0(0,ρ0) in ℋ0, that is, for any ε>0 and any bounded subset B0⊂ℋ0, there is a positive constant t0=t(B0,ρ0) such that
(3.5)S(t)B⊂B0for  anyt≥t0,u0,u1∈B0.Next, let us establish the existence of the bounded absorbing set inℋ1.Lemma 3.3.
Under the conditions (1.4), (1.5), (1.6), {S(t)}t≥0 has a bounded absorbing set ℬ≜Bℋ1(0,ρ1) in ℋ1, that is, for any ε>0 and any bounded subset B⊂ℋ1, there is a positive constant T=T(B,ρ1) such that
(3.6)S(t)B⊂Bfor anyt≥T,u0,u1∈B.Proof.
Take the scalar product inH of (1.1) with Av=Aut+σAu, we have
(3.7)12ddt(|Au|2+‖v‖2)+σ|Au|2+(α-σ)‖v‖2-σ(α-σ)(Au,v)+(φ(u),Av)=(f,Av).
For 0<σ≤σ0, σ0={α/4,λ1/2α}, by Hölder inequality, Poincaré inequality, and Cauchy inequality we have
(3.8)σ|Au|2+(α-σ)‖v‖2-σ(α-σ)(Au,v)≥σ|Au|2+(α-σ)‖v‖2-σ(α-σ)λ1|Au|‖v‖≥σ|Au|2+34α‖v‖2-σαλ1|Au|‖v‖≥σ|Au|2+34α‖v‖2-(σ2|Au|2+σα22λ1‖v‖2)≥σ2|Au|2+α2‖v‖2.
It follows from (1.5) that, for any ε>0, there exists a constant C1>0, such that
(3.9)|φ′(s)|≤ε|s|2+C1,∀s∈R.
Hence,
(3.10)|((φ(u),v))|=|∫Ωφ′(u)⋅∇u⋅∇v|dx≤∫Ω|φ′(u)|⋅|∇u|⋅|∇v|dx≤ε∫Ω|u|2⋅|∇u|⋅|∇v|dx+C1∫Ω|∇u|⋅|∇v|dx≤ε(∫Ω|u|4⋅|∇u|2dx)1/2⋅‖v‖+C1‖u‖⋅‖v‖≤ε(∫Ω|u|6dx)1/3⋅(∫Ω|∇u|6dx)1/6⋅‖v‖+C1‖u‖⋅‖v‖≤ε‖u‖2⋅C22⋅|Au|⋅‖v‖+α16‖v‖2+4C12α‖u‖2,
where C2 is the positive constant satisfying
(3.11)C2‖u‖2≥(∫Ω|u|6dx)1/3,C2|Au|≥(∫Ω|∇u|6dx)1/6,(3.12)(f,Av)=ddt(f,Au)+σ(f,Au)≤ddt(f,Au)+σ8|Au|2+2σ|f|2.
If (u0,v0) belongs to a bounded set B of ℋ1, then B is also bounded in ℋ0, and for t≥t0, by Lemma 3.2, we have
(3.13)‖u‖2+|ut|2≤ρ02;t0, ρ0 are given in Lemma 3.2 Choose
(3.14)0<ε2≤ασ16ρ04C24,
it follows from (3.10) that
(3.15)|((φ(u),v))|≤α8‖v‖2+4ε2α‖u‖4C24|Au|2+4C12α≤α8‖v‖2+σ4|Au|2+4C12αρ02,t≥t0.
Combining with (3.8), (3.12), and (3.15), by the Hölder inequality and the Young inequality, we deduce from that (3.7):
(3.16)ddt(|Au|2+‖v‖2-2(f,Au))+σ4|Au|2+α4|v|2≤4σ|f|2+8C12αρ02.
Let y=|Au-f|2+∥v∥2, from the above inequality, we can obtain
(3.17)dydt+σ8y≤dydt+σ8(|Au|2+‖v‖2+|f|2)+σ4|Au||f|=dydt+σ4(|Au|2+‖v‖2)-σ8(|Au|2+‖v‖2)+σ4|Au||f|+σ8|f|2≤4σ|f|2+8C12αρ02-σ8(|Au|2+‖v‖2)+σ8(|Au|2+|f|2)+σ8|f|2≤4σ|f|2+8C12αρ02+σ4|f|2-σ8‖v‖2≤σ4|f|2+4σ|f|2+8C12αρ02≤92σ|f|2+8C12αρ02.
Let C3=(9/2)σ|f|2+(8C12/α)ρ02. By the Gronwall lemma, we have
(3.18)y(t)≤y(t0)exp⁡(-σ8(t-t0))+8C3σ.
Defining ρ1′ by
(3.19)ρ1′2=8C3σ,
we see that
(3.20)limsup⁡t→∞y(t)≤ρ1′2,
and we conclude that
The ball ofℋ1, ℬ≜Bℋ1(0,ρ1), centered at (A-1f,0) of radius ρ1>ρ1′+ρ0, is absorbing in ℋ1 for the semigroup S(t),t≥0.We now give the property of compactness about the nonlinear operatorφ which will be needed in the proof of the condition (C).Lemma 3.4.
Assume thatφ∈C1(R,R) and φ:D(A)→V are defined by
(3.21)((φ(u),v))=∫Ωφ′(u)∇u∇vdx,
for all u∈D(A),v∈H01(Ω). Then, φ is continuous compact.Proof.
Let{um} be a bounded sequence in D(A). Without loss of generality, we assume that {um} weakly converges to u0 in D(A), since D(A) is reflexive. By the Sobolev embedding theorem, we know that H2(Ω)↪L∞⋂H1(Ω) and the embedding H2(Ω)↪H1(Ω) is compact in R3. Hence, we have that
(3.22)um⟶u0inH1.
Furthermore, there exists a constant C such that
(3.23)‖u0‖L∞⋂H1(Ω)≤C,‖um‖L∞⋂H1(Ω)≤C.
It is sufficient to prove that {φ(um)} converges to {φ(u0)} in V:
(3.24)‖φ(um)-φ(u0)‖=(∫Ω|φ′(um)∇(um)-φ′(u0)∇u0|2dx)1/2≤(∫Ω|φ′(um)|2|∇(um)-∇u0|2dx)1/2+(∫Ω|∇u0|2|φ′(um)-φ′(u0)|2dx)1/2.
On the one hand, for the first term in (3.24), combining with (3.23) and the continuity of φ′(·), we have
(3.25)∫Ω|φ′(um)|2|∇(um)-∇u0|2dx≤C‖um‖L∞(Ω)‖∇(um)-∇u0‖2.
On the other hand, for the second term in (3.24), using the continuity of φ′(·),
(3.26)∫Ω|∇u0|2|φ′(um)-φ′(u0)|2dx⟶0,asm⟶0,
follows immediately by dominated convergence theorem.
Also, considering (3.22), passing to the limit in (3.24), we can obtain
(3.27)lim⁡m→∞‖φ(um)-φ(u0)‖=0.
This completes the proof.Lemma 3.5.
Suppose the conditions (1.4), (1.5), (1.6) hold, the solution semigroup {S(t)}t≥0 of the problem (1.1)–(1.3) satisfies the condition (C) in ℋ1.Proof.
Let{ωi} be an orthonormal basis of L2(Ω) which consists of eigenvalues of A. The corresponding eigenvalues are denoted by {λj}j=1∞:
(3.28)0<λ1≤λ2≤λ3≤⋯≤λj≤⋯,λj⟶∞
with
(3.29)Aωi=λiωi,∀i∈N.
Let Vm=  span{ω1,…,ωm} in V and let Pm:V→Vm be an orthogonal projector. We write
(3.30)u=Pmu+(I-Pm)u≜u1+u2.
Taking the scalar product of (1.1) in H with Av2=Au2t+σAu2, we find
(3.31)12ddt(|Au2|2+‖v2‖2)+σ|Au2|2+(α-σ)‖v2‖2-σ(α-σ)(Au2,v2)+(φ(u),Av2)=(f,Av2).
Choose 0<σ≤σ0, similar to (3.8), we have
(3.32)σ|Au2|2+(α-σ)‖v2‖2-σ(α-σ)(Au2,v2)≥σ2|Au2|2+α2‖v2‖2.
Since f∈L2(Ω), φ:D(A)→V is compact by Lemma 3.4, for any ε>0, there exists some m such that
(3.33)|(I-Pm)f|H≤ε,(3.34)‖(I-Pm)φ(u)‖V≤ε,∀u∈B1(0,ρ1),
where ρ1 is given by Lemma 3.2.
By exploiting the Hölder inequality and Cauchy inequality, we have(3.35)(φ(u),Av2)≤‖(I-Pm)φ‖‖v2‖≤ε∥v2∥≤α4‖v2‖2+ε2α,(3.36)(f,Av)=ddt(f,Au)+σ(f,Au)≤ddt(f,Au)+σ4|Au|2+σ|f|2.
By (3.33) and (3.36), we have
(3.37)(f,Av2)=ddt(f2,Au2)+σ(f,Au2)≤ddt(f2,Au2)+σ4|Au2|2+σ|f2|2≤ddt(f2,Au2)+σ4|Au2|2+σε2,
where f2≜(I-Pm)f.
Hence, combining with (3.32), (3.35), and (3.37), we obtain from (3.31) that
(3.38)ddt(|Au2|2+‖v2‖2-2(f2,Au2))+σ2(|Au2|2+‖v2‖2)≤(2σ+2α)ε2.
Let y=|Au2-f2|2+∥v2∥2, from the above inequality, similar to (3.17), we can obtain
(3.39)dydt+σ4y≤dydt+σ4(|Au2|2+‖v2‖2+|f2|2)+σ2|Au2||f2|≤(5σ2+2α)ε2.
Let C4=5σ/2+2/α. By the Gronwall lemma, we have
(3.40)y(t)≤y(t1)exp⁡(-σ4(t-t1))+4C4ε2σ.
Choosing t2=t1+(4/σ)ln(ρ12/ε2), it follows that
(3.41)y(t)≤(1+4C4σ)ε2,t≥t2.
That is,
(3.42)|Au2-f2|2+‖v2‖2≤C̃ε2,
for all t≥t2, where C̃=(1+4C4/σ).
Thus we complete the proof.We are now in a position to state our main results as follows.Theorem 3.6.
Under the conditions (1.4), (1.5), (1.6), problem (1.1)–(1.3) has a global attractor 𝒜1 in ℋ1; it attracts all bounded subsets of ℋ1 with respect to the norm of ℋ1.Proof.
By Lemmas3.1, 3.3, and 3.5, the conditions of Theorem 2.5 are satisfied. The proof is complete.Corollary 3.7.
The global attractor𝒜0 in ℋ0 is coincides with 𝒜1 in ℋ1,thatis,𝒜0=𝒜1.Proof.
By Theorem2.6, 𝒜0 is a bounded set of ℋ1, combining with Theorem 3.6, we can easily get 𝒜0=𝒜1.



---

*Source: 469382-2012-06-13.xml*