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

This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma (u){\mathrm{\partial }}_{t}u$ 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 $H_0^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ is obtained. Moreover, the regularity of the global attractor is established, implying that it is a bounded subset of $(H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega }))\times H_0^1(\mathrm{\Omega })$.