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

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 $(H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega)$.