# American Institute of Mathematical Sciences

January  2016, 21(1): 205-225. doi: 10.3934/dcdsb.2016.21.205

## Attractors for wave equations with nonlinear damping on time-dependent space

 1 School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China 2 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 3 Department of Mathematics, Nanjing University, Nanjing 210093

Received  January 2015 Revised  July 2015 Published  November 2015

In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation \begin{eqnarray*} \varepsilon(t) u_{tt}+g(u_{t})-\Delta u+\varphi (u)=f \end{eqnarray*} with Dirichlet boundary condition, in which, the coefficient $\varepsilon$ depends explicitly on time, the damping $g$ is nonlinear and the nonlinearity $\varphi$ has a critical growth. Spirited by this concrete problem, we establish a sufficient and necessary condition for the existence of attractors on time-dependent spaces, which is equivalent to that provided by M. Conti et al.[10]. Furthermore, we give a technical method for verifying compactness of the process via contractive functions. Finally, by the new framework, we obtain the existence of the time-dependent attractors for the wave equations with nonlinear damping.
Citation: Fengjuan Meng, Meihua Yang, Chengkui Zhong. Attractors for wave equations with nonlinear damping on time-dependent space. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 205-225. doi: 10.3934/dcdsb.2016.21.205
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