Well-posedness and global attractor for wave equation with displacement dependent 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 displacement-dependent damping and super-cubic nonlinearity, extending previous results with refined estimates.

Contribution

It establishes well-posedness and the existence of a regular global attractor for the wave equation with nonlinear damping and super-cubic nonlinearity, using refined a priori estimates.

Findings

Well-posedness of weak solutions is proven.

Existence of a bounded global attractor in phase space is shown.

The global attractor has higher regularity, lying in a more regular function space.

Abstract

This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity exponents with $\sigma(u)$ and $f(u)$, the well-posedness of the weak solution is established. Furthermore, the existence of a global attractor in the naturally phase space $H^1_0(\Omega)\times L^2(\Omega)$ is obtained. Moreover, the regularity of the global attractor is established, implying that it is a bounded subset of $(H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega)$.