# Strong Exponential Attractors for Weakly Damped Semilinear Wave Equations

Authors: Cuncai Liu; Fengjuan Meng; Chang Zhang
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we investigate the longtime dynamics for the damped wave equation in a bounded smooth domain ofℝ3. The exponential attractor is investigated in a strong energy space for the case of subquintic nonlinearity, which is based on the recent extension of the Strichartz estimate for the case of a bounded domain. The results obtained complete some previous works.


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## Body

## 1. Introduction

LetΩ⊂ℝ3 be a bounded domain with smooth boundary. Given γ>0, we consider the following weakly damped wave equation:(1)utt+γut−Δu+fu=g,x∈Ω,t>0,ux,0=u0,utx,0=u1,x∈Ω,ux,t=0,x∈∂Ω,t>0.Here,g∈L2Ω is independent of time and the nonlinear function f∈C3ℝ with f0=0 and satisfies the following conditions:(2)f‴s≤c11+sp−3,liminfs⟶∞fss>−λ1,where 3≤p<5 and λ1>0 is the first eigenvalue of −Δ on Ω with Dirichlet boundary condition.Denote by⋅,⋅ and ⋅ the inner product and norm in L2Ω. For the sake of convenience, we define Hilbert spaces ℋ=H01Ω×L2Ω, ℋ1=H2Ω∩H01Ω×H01Ω, and ℋ2=H3Ω∩H01Ω×H2Ω∩H01Ω.Dissipative wave equations arise as an evolutionary mathematical model in various systems for the relevant physical application including electrodynamics, quantum mechanics, and nonlinear elasticity (see e.g., [1]). The longtime behavior of problem (1) is of a big permanent interest (see e.g. [2–5]), which depends strongly on the growth rate of the nonlinearity f. For a long time, in a bounded smooth domain of ℝ3, the global well-posedness for problem (1) holds only in the case of subcubic or cubic nonlinearity (that is, the exponent p in 1.2 less or equals to 3), in which the uniqueness is verified by the technology of Sobolev embedding in general. Therefore, the cubic growth rate of nonlinearity had been considered as a critical one for the case of 3-D bounded domain. Consequently, the existence of global attractors for the weakly damped wave equations in the natural energy space ℋ as well as its regularities only had been known for the case p≤3, that is, the nonlinear term f at most can be cubic growth in 3-D bounded domain (see [2, 3, 5, 6] and the references therein).The case of supercubic growth rate is a bit more complicated since the uniqueness of energy weak solutions is unknown for the supercubic case (e.g., see [7]). However, this problem can be overcome using slightly more regular solutions than the energy ones. These solutions are the so-called Shatah–Struwe solutions. Recently, with the help of the suitable versions of Strichartz estimates, the global well-posedness of quintic wave equation in terms of Shatah–Struwe solutions in 3-D smooth bounded domains has been obtained in [8–10]. Based on the fact that the Strichartz estimates can be generalized to weakly damped wave equation, Kalantarov et al. [11] obtained the well-posedness of weakly damped wave equation in the case of quintic nonlinearity in bounded domain of ℝ3, where the solution is the weak solution with extra regularity named Shatah–Struwe solution; moreover, they also considered the global attractor for the case of quintic nonlinearity.Considering the strong solution, in 1983, Babin and Vishik [3] investigated the damped subcubic wave equation in ℋ. For seeking a compact absorbing set in ℋ, they requested the initial data belonging to ℋ1, and they proved the existence of the maximal ℋ1,ℋ attractor for problem (1). Then, Ladyzhenskaya [12] obtained the ℋ1 attractor for cubic nonlinearity. In [13], the second author verified the ℋ1 attractor via a different approach. Recently, the authors [14] have obtained the strong global attractor for subquintic nonlinearity.As mentioned by many authors, the global attractor, however, does not provide an actual control of the convergence rate of trajectories and it can be sensitive to perturbations. In order to overcome these difficulties, Eden et al. introduced the notion of exponential attractor (see [15, 16]). In contrary to the global attractor, the exponential attractor is not unique, and it is only semi-invariant. However, it has the advantage of being stable with respect to perturbations, attracts the trajectories exponentially, and has finite fractal dimension. For the strongly damped wave equation, in the subcritical case, Pata and Squassina [17] established an exponential attractor of optimal regularity by using the bootstrap argument; in the critical case, Sun and Yang [18, 19] obtained the exponential attractors based on some asymptotic regularity of the solution. For the weakly damped wave equation, Eden et al. [20] and Grasselli and Pata [21] investigated the exponential attractor for problem (1) with cubic nonlinearity in ℋ1 and ℋ, respectively. For the supercubic nonlinearity, in [22], we established the exponential attractor in ℋ. However, to the best of our knowledge, there is no result on exponential attractor for problem (1) in ℋ1.In the present paper, as a continuation of our previous works [14, 22], we concentrate on the existence of exponential attractor in the subquintic but supercubic case and establish an exponential attractor in the strong topology space based on the Shatah–Struwe solution.Our main result can be stated as follows.Theorem 1.
Letg∈L2Ω and f satisfy (2). Then, the solution semigroup St acting on ℋ1 possesses an exponential attractor A1 which is compact in ℋ1, satisfying the following conditions:(i)
A1 is positive invariant, i.e., StA1⊂A1 for all t≥0.(ii)
dimFA1<∞, i.e., A1 has finite fractal dimension.(iii)
There exist a monotone increasing functionQ1⋅ and positive constant μ such that for any bounded set B⊂ℋ1,(3)distℋ1StB,A1≤Q1Bℋ1e−μt,for allt≥0.The above theorem will be proved in Section3. Before that, the preliminary things, including some notations, the well-posedness, and the global attractor of the system are discussed in Section 2.Throughout the rest of the paper, denote byC any positive constants which may be different from line to line even in the same line; we also denote the different positive constants by Ci,i∈ℕ, for special differentiation.


## 2. Well-Posedness and Global Attractor

Firstly, we recall the definitions about the weak solution and strong solution of problem (1) in [11], which will be used to state our results.Definition 1.
(see [11]). For any T>0, a function ut is a weak solution of problem (1) if u,ut∈L∞0,T;ℋ and equation (1) is satisfied in the sense of distribution, i.e.,(4)−∫0Tut,ϕtdt+γ∫0Tut,ϕdt+∫0T∇u,∇ϕdt+∫0Tfu,ϕdt=∫0Tg,ϕdt,for any ϕ∈C0∞0,T×Ω. Furthermore, ut is a strong solution if u,ut∈L∞0,T;ℋ1.
For the well-posedness of strong solution, we have the following result.Theorem 2 (see [14]).
Letg∈L2Ω and f satisfy (2). Then, for every u0,u1∈ℋ1, there exists a unique global strong solution ut of problem (1) with energy estimate(5)ut,uttℋ1≤Qu0,u1ℋ1,g,where monotone function Q is independent of u and t. Moreover, for any two strong solutions u1t and u2t, there exists a positive constant C1 depending on the ℋ1-norm of u1t and u2t such that(6)u¯t,u¯ttℋ1≤eC1Tu¯0,u¯t0ℋ1,t∈0,T,
whereu¯t=u1t−u2t.
Therefore, the strong solution generates aC0 operator semigroup Stt≥0 in ℋ1 as follows:(7)Stu0,u1=ut,utt,which is continuous in ℋ1.Finally, we state the existence of global attractor in strong energy space which will be used in Lemma3.Theorem 3 (see [14]).
Letg∈L2Ω and f satisfy (2). Then, the solution semigroup St associated with problem (1) possesses a global attractor A in the space ℋ1.


## 3. Proof of Theorem1

There are two different approaches to find exponential attractors. The original one (cf. [15, 16]) relies on the proof that the semigroup St satisfies squeezing property, and the following approach (cf. [16, 17, 23]) is established via decomposition of semigroup, which is easier to verify for damped wave equation.Lemma 1.
LetX0 and X1 be two Banach spaces such that X1 is compactly embedded into X0 and Stt≥0 is a semigroup on X0. Assume that there exists a bounded positively invariant X⊂X0 and a time t∗>0 such that the following hold:(i)
The map(8)t,z0⟼Stz0:0,t∗×X⟶Xis Lipschitz continuous (with the metric inherited fromX0).(ii)
The mapSt∗:X⟶X admits a decomposition of the form(9)St∗=S0+S1,S0:X⟶X0,S1:X⟶X1,whereS0 and S1 satisfy the conditions(10)S0z1−S0z2X0≤14z1−z0X0,∀z1,z2∈X,S1z1−S1z2X1≤C∗z1−z2X0,∀z1,z2∈X,for someC∗>0.Then, the local dynamical systemX,St possesses an exponential attractor.In order to verify the local exponential attractor inX is an exponential attractor in ℋ1, the following transitivity of exponential attraction is necessary.Lemma 2 (see [24]).
Letℳ,d be a metric space and St be a semigroup acting on this space such that(11)dStm1,Stm2≤L0eθ0tdm1,m2,for some nonnegative constants L0 and θ0. Furthermore, we assume that there exist three subsets M1,M2,M3⊂ℳ such that(12)distℳStM1,M2≤L1e−θ1t,distℳStM2,M3≤L2e−θ2t,for some θ1,θ2>0 and L1,L2≥0. Then it follows that(13)distℳStM1,M3≤Le−θt,where L=L0L1+L2 and θ=θ1θ2/θ0+θ1+θ2.We will verify all conditions of Lemma1 step by step.Step 1.
Construct a bounded positively invariant set.
Decompose the solutionut as(14)ut=−Δ−1g+vt+wt,where vt solves the linear problem(15)vtt+γvt−Δv=0,v0=u0+Δ−1g,vt0=u1,and the remainder wt satisfies(16)wtt+γwt−Δw=−fu,w0=0,wt0=0.
It is convenient to denote(17)Svtz=vt,vtt,Swtz=wt,wtt,where z=u0,u1. For equation (15), it is easy to check that the solution is exponentially decaying, namely,(18)Svtzℋ1≤C3e−ρtzℋ1+gholds for some ρ>0. From [14, Lemma 4.2], we have the following regularity estimate for Swt:(19)Swtzℋ2≤C∫0teαs−tfusH2ds≤C4zℋ1,g,with α>0 small enough.Lemma 3.
Let the assumptions in Theorem1 hold. Then, there exist a compact positive invariant set ℬ1 in ℋ1, positive constant μ, and a monotone increasing function Q2⋅ such that for any bounded set B⊂ℋ1,(20)distℋ1StB,ℬ1≤Q2Bℋ1e−μtfor allt≥0.Moreover, the setℬ1+Δ−1g,0 is bounded in ℋ2.Proof.
After vanishing thev-component, we can obtain a “perturbed positive trajectory.” And the original is asymptotically close to positive trajectory exponentially. Therefore, a collection of perturbed positive trajectories will be a promising candidate for an exponentially attracting set. To carry out this idea, denote K0=−Δ−1g,0+∪t≥0SwtN¯ℋ1, where N=z∈ℋ1:distℋ1z,A<1 is a neighborhood of the global attractor A. We will show that K0 is an exponentially attracting set. Notice that N is an absorbing set. Hence, for any given bounded set B⊂ℋ1, there exists T=TBℋ1 such that StB⊂N, for all t≥T. Furthermore, for t≥T, we have(21)distℋ1StB,K0≤distℋ1St−TSTB,−Δ−1g,0+Swt−TN≤distℋ1St−TN,−Δ−1g,0+Swt−TN≤Svt−TNℋ1≤C3e−ρt−TNℋ1+g.
On the other hand, by inequality (5), we have the uniform estimate(22)distℋ1StB,K0≤C5,t≥0,for some C5 depending only on Bℋ1. Combining the two above inequalities, we obtain(23)distℋ1StB,K0≤C5eρT+C3eρTNℋ1+ge−ρt,t≥0.
Then, we define a positive invariant set byK1=∪t≥0StK0¯V, which satisfies(24)distℋ1StB,K1≤C5+C3eρTNℋ1+ge−ρt,t≥0.
It remains to show thatK1+Δ−1g,0 is bounded in ℋ2. Inequality (19) implies SwtK0 and K0+Δ−1g,0 are both bounded in ℋ2. Because StK0⊂−Δ−1g,0+SvtK0+SwtK0, we will complete our proof if we can show SvtK0 is bounded (uniformly in t≥0) in ℋ2. Note that Svtu0,u1 is generated by linear problem(25)vtt+γvt−Δv=0,v0=u0+Δ−1g,vt0=u1,and it is clear that Sv⋅u0,u1∈L∞ℝ+;ℋ2 if u0+Δ−1g,u1∈ℋ2. Thus, K0+Δ−1g,0⊂ℋ2 implies SvtK0 is bounded in ℋ2. Therefore, by setting ℬ1=K1, the proof is completed.Step 2.
We will verify asymptotic regularity and Lipschitz continuity.Lemma 4.
Let the assumptions in Theorem1 hold. Then, there exists a time t∗>0 such that the following properties hold:(i)
We have estimates(26)Svt∗z1−Svt∗z2ℋ1≤14z1−z2ℋ1,for allz1,z2∈ℬ1,Swt∗z1−Swt∗z2ℋ2≤Kz1−z2ℋ1for allz1,z2∈ℬ1,where the constantK depends only on t∗ and ℬ1ℋ1.(ii)
The mappingt,z⟼Stz is Lipschitz continuous in ℋ1-norm on 0,t∗×ℬ1.Proof.
(i) For anyz1=u01,u11,z2=u02,u12∈ℬ1, set v¯t,v¯tt=Svtz1−Svtz2 and w¯t,w¯tt=Swtz1−Swtz2. Then, v¯t,v¯tt and w¯t,w¯tt solve the following equations, respectively:(27)v¯tt+γv¯t−Δv¯=0,v¯0=u01−u02,v¯t0=u11−u12,(28)w¯tt+γw¯t−Δw¯=fu2−fu1,w¯0=0,w¯t0=0.
From equation (27), we can get exponential decay estimate v¯t,v¯ttℋ1≤C3e−ρtz1−z2ℋ1. Hence, by setting t∗=1/ρln4C3, we have(29)v¯t,v¯ttℋ1≤14z1−z2ℋ1.
Taking inner product of (28) with Δ2w¯t and integrating on 0,t∗, we can obtain(30)w¯t,w¯ttℋ2≤C∫0t∗fu2s−fu1sH2∩H012ds1/2.
Since(31)fu2−fu1H2∩H012=Δfu2−fu1=Δ∫01f′u1−τu¯u¯dτ=∫01f‴u1−τu¯∇u1−τu¯2u¯+2f″u1−τu¯∇u1−τu¯∇u¯+f′u1−τu¯Δu¯dτ≤C1+u1H2p−1+u2H2p−1Δu¯,we have(32)w¯t,w¯ttℋ2≤C∗z1−z2ℋ1,for some C∗ depending on ℬ1ℋ1 and t∗.
(ii) Forz1,z2∈ℬ1 and t1,t2∈0,t∗, we have(33)St1z1−St2z2ℋ1≤St1z1−St1z2ℋ1+St1z2−St2z2ℋ1.
The first term is handled by estimate (6). Recall that ℬ1 is invariant, and thus S⋅ℬ1+Δ−1g,0∈L∞0,∞;ℋ2. Substituting the ℋ2 regularity result into equation (1), we have d/dtS⋅ℬ1∈L∞0,∞;ℋ1. This yields that for the second term in the right hand of (33), we have(34)St1z2−St2z2ℋ1≤∫t1t2ddtSsz2ℋ1ds≤ddtSsz2L∞0,t∗;ℋ1t1−t2.
Therefore, applying Lemma1, we can prove the existence of an exponential attractor A1 for Stt≥0 in ℬ1 immediately. From the construction of ℬ1, ℬ1 exponentially attracts all bounded subsets of the whole phase-space ℋ1. Then, the transitivity property of exponential attraction implies that A1 attracts all bounded subsets of the whole phase-space ℋ1 exponentially, i.e., A1 is an exponential attractor in ℋ1.



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