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

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 thetime-dependent attractors for the wave equations with nonlinear damping.