  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
$\partial_{tt}u + \beta(t)\partial_tu = \mathcal{L}u(x,t) + f(u)$ 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.
