Blow--up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources

Abstract

The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u=0 &\text{on $(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t)=g(x,u)\qquad &\text{on $(0,\infty)\times \Gamma_1$,}\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & \text{in $\overline{\Omega}$,} \end{cases}$$ where $\Omega$ is a bounded open $C^1$ subset of $\mathbb{R}^N$, $N\ge 2$, $\Gamma=\partial\Omega$, $(\Gamma_0,\Gamma_1)$ is a partition of $\Gamma$, $\Gamma_1\not=\emptyset$ being relatively open in $\Gamma$, $\Delta_\Gamma$ denotes the Laplace--Beltrami operator on $\Gamma$, $\nu$ is the outward normal to $\Omega$, and the terms $P$ and $Q$ represent nonlinear damping terms, while $f$ and $g$ are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well--posedness.