Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

Abstract

We consider the global existence of strong solution $u$, corresponding to a class of fully nonlinear wave equations with strongly damped terms $u_{tt}-k\mathrm{\Delta }{u}_t=f(x,\mathrm{\Delta }u)+g(x,u,Du,D^{2}u)$ in a bounded and smooth domain $\mathrm{\Omega }$ in $R^n$, where $f(x,\mathrm{\Delta }u)$ is a given monotone in $\mathrm{\Delta }u$ nonlinearity satisfying some dissipativity and growth restrictions and $g(x,u,Du,D^{2}u)$ is in a sense subordinated to $f(x,\mathrm{\Delta }u)$. By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution $u\in L_{\text{l}\text{o}\text{c}}^{\infty }((0,\infty ),W^{2,p}(\mathrm{\Omega })\cap W_0^{1,p}(\mathrm{\Omega }))$.