Project Euclid

We study the global well-posedness of the nonlinear wave equation $$ u_{tt} - \Delta u - \Delta _p u_t = f(u) $$ in a bounded domain ${\Omega} \subset \mathbb{R}^n$ with Dirichlét boundary conditions. The nonlinearity $f(u)$ represents a strong source which is allowed to have a supercritical exponent; i.e., the Nemytski operator $f(u)$ is not locally Lipschitz from $H^1_0({\Omega})$ into $L^2({\Omega})$. The nonlinear term $- \Delta _p u_t $ is a strong damping where the $-\Delta _p$ denotes the p-Laplacian (defined below). Under suitable assumptions on the parameters and with careful analysis involving the theory of monotone operators, we prove the existence and uniqueness of a local weak solution. Also, such a unique solution depends continuously on the initial data from the finite energy space. In addition, we prove that weak solutions are global, provided the exponent of the damping term dominates the exponent of the source.