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

Abstract

This work investigates the semilinear wave equation featuring the displacement dependent term $\ddot{I}ƒ(u){\partial }_{t}u$ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity exponents with $\ddot{I}ƒ(u)$ and $f(u)$, the well-posedness of the weak solution is established. Furthermore, the existence of a global attractor in the naturally phase space $H_0^1(\hat{I}\text{c◯})\times L^2(\hat{I}\text{c◯})$ is obtained. Moreover, the regularity of the global attractor is established, implying that it is a bounded subset of $(H^2(\hat{I}\text{c◯})\cap H_0^1(\hat{I}\text{c◯}))\times H_0^1(\hat{I}\text{c◯})$.