Well-posedness and global attractor for wave equation with nonlinear damping and super-cubic nonlinearity

TL;DR

This paper proves the well-posedness and existence of a regular global attractor for a wave equation with nonlinear damping and super-cubic nonlinearity, broadening understanding of long-term dynamics in such systems.

Contribution

It establishes well-posedness and the existence of a regular global attractor for a class of wave equations with nonlinear damping and super-cubic nonlinearities, extending previous results.

Findings

Well-posedness of weak solutions in broader exponent ranges

Existence of a regular global attractor in phase space

Global attractor is bounded in higher regularity spaces

Abstract

This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing a priori space-time estimates. Moreover, the existence of a global attractor in the phase space $H^1_0(\Omega)\times L^2(\Omega)$ is obtained. Furthermore, it is proved that this global attractor is regular, implying that it is a bounded subset of $(H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega)$.