Exponential attractors for the strongly damped wave equations

Abstract

For the strongly damped wave equation with critical nonlinearity, we first show the existence of a $(H_0^1\left(\Omega \right)\times L^2\left(\Omega \right),H_0^1\left(\Omega \right)\times H_0^1\left(\Omega \right))$-global attractor when the external forcing $g\in H^{-1}$; then we prove that for each $T>0$ fixed, there is a bounded (in $H^2\times H^1$) set which attracts exponentially every $H^1\times L^2$-bounded set w.r.t. the stronger $H^1\times H^1$-norm for all $t⩾T$ and has finite fractal dimension in $H_0^1\left(\Omega \right)\times H_0^1\left(\Omega \right)$ for the case $g\in L^2\left(\Omega \right)$.

Introduction

We consider the following strongly damped wave equation on a bounded domain $\Omega \subset {\mathbb{R}}^3$ with smooth boundary $\partial \Omega$:

$$\{\begin{array}{c}{\partial }_{tt}u-\Delta {\partial }_{t}u-\Delta u+f\left(u\right)=g\left(x\right)\text{,}\\ (u\left(0\right),{\partial }_{t}u\left(0\right))=(u_0,v_0)\text{,}\\ u{|}_{\partial \Omega }=0\text{,}\end{array}$$

where $f\in {\mathcal{C}}^1\left(\mathbb{R}\right)$ with $f\left(0\right)=0$ and satisfies the following conditions:

growth condition

$$\left|f^{\prime }\left(s\right)\right|⩽C(1+{\left|s\right|}^p)\text{for all}s\in \mathbb{R}\text{,}$$

dissipation condition

$$\underset{\left|s\right|\to \infty }{lim\; inf}\frac{f\left(s\right)}{s}>-{\lambda }_1\text{,}$$

where $0⩽p⩽4$ and ${\lambda }_1$ is the first eigenvalue of $-\Delta$ on $H_0^1\left(\Omega \right)$.

Denoting

$$\mathcal{H}=H_0^1\left(\Omega \right)\times L^2\left(\Omega \right)\text{,}\mathcal{V}=H_0^1\left(\Omega \right)\times H_0^1\left(\Omega \right)\text{,}$$

and $\Vert \cdot \Vert$ for the norm of $L^2\left(\Omega \right)$.

It is known (e.g., see [1], [2]) that under the conditions (1.2), (1.3), Eq. (1.1) generates a $C_0$-semigroup ${\left\{S\left(t\right)\right\}}_{t⩾0}$ in the natural energy phase space $\mathcal{H}$. The asymptotic behavior of solutions to Eq. (1.1) has been the object of extensive studies via attractors, see [2], [3], [4], [5], [6], [7], [8] and the references therein.

Let us recall some recent relevant researches in this area.

About the existence of global attractor, in the subcritical case (i.e., $p<4$) it can be obtained by the standard theory of dynamic systems since $H_0^1\left(\Omega \right)\hookrightarrow L^q\left(\Omega \right)$ is compact for every $q<6$; in the critical case (i.e., $p=4$), it has been proven in [2] for $g=0$, and in [5] for $g\in H^{-1}$ recently.

About the regularity of attractor, for the subcritical case, the authors in [5] have proved that the global attractor is bounded in $H^2\times H^1$, and based on such regularity results, by the use of the abstract framework developed in Efendiev, Miranville and Zelik [9], they obtained further the existence of exponential attractor. For the critical case, Pata and Zelik [6] have proved that the global attractor is bounded in $H^2\times H^2$ as $g\in L^2\left(\Omega \right)$ when the nonlinearity $f(\cdot )$ satisfies ${lim\; inf}_{\left|s\right|\to \infty }{f}^{\prime }\left(s\right)>-{\lambda }_1$, $\forall s\in \mathbb{R}$ and the authors in [6] also pointed out further that one can prove the regularity of the attractor when $f(\cdot )$ only satisfies (1.2), (1.3), which have been realized recently in [10], [11], [12].

Since Eq. (1.1) contains the strong damping term $\Delta {\partial }_{t}u$, which brings many advantages for us to consider the long time behavior, especially in considering the attractors. For example, for any $(u_0,v_0)\in \mathcal{H}$ and $[t_1,t_2]\subset {\mathbb{R}}^{+}$, we know that the corresponding solution $(u\left(t\right),u_t\left(t\right))=S\left(t\right)(u_0,v_0)$ satisfies $u_t\left(t\right)\in L^2(t_1,t_2;H_0^1\left(\Omega \right))$, which somehow shows that there is some kind of regularity about the time derivative term $u_t$. Indeed, Pata and Zelik [6] have proven the following crucial regularity results about $u_t$, here we recall them as follows (see Lemmas 3.5 and 3.6 of [6], their methods are applicable for $g\in H^{-1}$ and $f$ satisfies (1.3)):

Lemma 1.1 [6]

Under conditions(1.2), (1.3), for every $t>0$ , the following estimate holds

$$min\{1,t\}{\Vert \nabla {\partial }_{t}u\left(t\right)\Vert }^2+min\{1,t^2\}{\Vert {\partial }_{tt}u\left(t\right)\Vert }^2⩽Q({\Vert (u_0,v_0)\Vert }_{\mathcal{H}}+{\Vert g\Vert }_{H^{-1}})\text{,}$$

where $Q(\cdot )$ is a nondecreasing function on $[0,\infty )$ , and $(u\left(t\right),u_t\left(t\right))$ is the solution corresponding to the initial data $(u_0,v_0)\in \mathcal{H}$ .

Hence, a natural and interesting problem is to discuss the attraction in a slightly stronger space $\mathcal{V}$, which will reflect the strongly damped properties of $\Delta {\partial }_{t}u$ to some extent, in other words, for the second ingredient $u_t\left(t\right)$ of solution $(u\left(t\right),u_t\left(t\right))$, it should behave like the solution of a parabolic equation.

In this paper, we characterize some stronger (in $\mathcal{V}$-topology) asymptotic properties of (1.1) by the concepts of global attractor and exponential attractor:

• after some preliminaries in Section 2, we first prove the existence of compact $(\mathcal{H},\mathcal{V})$-global attractor when $g\in H^{-1}$ in Section 3 (i.e., attracts every $\mathcal{H}$-bounded set w.r.t. the $\mathcal{V}$-norm), see Theorem 3.7, the necessary asymptotic compactness is obtained by a different decomposition (3.2_η)–(3.3_η) of (1.1);

• then, in Section 4, we discuss the $(\mathcal{H},\mathcal{V})$-exponential attraction when $g\in L^2\left(\Omega \right)$. Since there is a singularity at $t=0$ for the solution of (1.1) (for example, from Lemma 1.1, for any initial data $(u_0,v_0)\in H_0^1\left(\Omega \right)\times L^2\left(\Omega \right)$, we know that the second ingredient $u_t\left(t\right)$ of the corresponding solution $(u\left(t\right),u_t\left(t\right))$ belongs to $H_0^1\left(\Omega \right)$ whenever $t>0$, even $v_0$ only belongs to $L^2\setminus H_0^1$), we cannot find in general that there is a bounded subset ${\mathcal{E}}^{\prime }\subset \mathcal{V}$ such that dist${}_{\mathcal{V}}(S\left(t\right)B,{\mathcal{E}}^{\prime })⩽C_B{e}^{-kt}$ for all $t⩾0$ and all $B\in \mathcal{H}$. But, we can establish the exponential attraction when $t⩾T$ for each $T>0$. Hence, we take a slightly modified definition of exponential attractor: we first prove that for each fixed $T>0$, there is a ${(\mathcal{H},\mathcal{V})}_T$-exponential attractor (see Definition 4.7 and Lemma 4.8), which has finite fractal dimension in $\mathcal{V}$ and attracts exponentially any bounded (in $\mathcal{H}$) set with respect to $\mathcal{V}$-norm for all $t⩾T$; then we prove the main result of this paper, Theorem 4.1.