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

Abstract

In this paper, we consider the long time behavior of the solution for the following nonlinear damped wave equation ε(t)utt + g(ut) - δu + φ(u) = f with Dirichlet boundary condition, in which, the coefficient " depends explic- itly on time, the damping g is nonlinear and the nonlinearity ' has a critical growth. Spirited by this concrete problem, we establish a sufficient and neces- sary 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.