Wave equation with nonlinear damping and super-cubic nonlinearity

This study investigates a semilinear wave equation characterized by nonlinear damping $g\left(u_t\right)$ and nonlinearity $f\left(u\right)$. 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 $H_0^1\left(\Omega \right)\times L^2\left(\Omega \right)$ is obtained. Furthermore, it is proved that this global attractor is regular, implying that it is a bounded subset of $\left(H^2\right(\Omega )\cap H_0^1(\Omega \left)\right)\times H_0^1\left(\Omega \right)$.