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

TL;DR

This paper investigates the long-term behavior of solutions to a degenerate wave equation with time-dependent damping, establishing well-posedness and the existence of pullback attractors under certain conditions.

Contribution

It introduces the analysis of pullback attractors for a critical degenerate wave equation with time-dependent damping, extending the understanding of its long-term dynamics.

Findings

Proved local and global well-posedness of solutions.

Established existence of pullback attractors.

Analyzed effects of critical nonlinear growth and damping restrictions.

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