Pullback Attractors for a Critical Degenerate Wave Equation with Time-dependent Damping

Abstract

The aim of this paper is to analyze the long-time dynamical behavior of the solution for a degenerate wave equation with time-dependent damping term ${\partial }_{tt}u+\beta (t){\partial }_{t}u=\mathcal{L}u(x,t)+f(u)$ on a bounded domain $\Omega \subset {\mathbb{R}}^N$ with Dirichlet boundary conditions. Under some restrictions on $\beta (t)$ and critical growth restrictions on the nonlinear term $f$, we will prove the local and global well-posedness of the solution and derive the existence of a pullback attractor for the process associated with the degenerate damped hyperbolic problem.