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 $H_0^1(\hat{I}\text{c◯})\times L^2(\hat{I}\text{c◯})$ is obtained. Furthermore, it is proved that this global attractor is regular, implying that it is a bounded subset of $(H^2(\hat{I}\text{c◯})\cap H_0^1(\hat{I}\text{c◯}))\times H_0^1(\hat{I}\text{c◯})$.

37 pages, 1 figure