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

The aim of this paper is to give global nonexistence and blow–up results for the problem $$\{\begin{array}{ll}{u}_{tt}-\mathrm{\Delta }u+P(x,u_t)=f(x,u)& \text{in }(0,\mathrm{\infty })\times \mathrm{\Omega }\text{,}\\ u=0& \text{on }(0,\mathrm{\infty })\times {\mathrm{\Gamma }}_0\text{,}\\ u_{tt}+{\mathrm{\partial }}_{\nu }u-{\mathrm{\Delta }}_{\mathrm{\Gamma }}u+Q(x,u_t)=g(x,u)& \text{on }(0,\mathrm{\infty })\times {\mathrm{\Gamma }}_1\text{,}\\ u(0,x)=u_0(x),u_t(0,x)=u_1(x)& \text{in }\bar{\mathrm{\Omega }}\text{,}\end{array}$$ where $\mathrm{\Omega }$ is a bounded open $C^1$ subset of ${\mathbb{R}}^N$, $N\ge 2$, $\mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$, $({\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ is a partition of $\mathrm{\Gamma }$, ${\mathrm{\Gamma }}_1≠\mathrm{\varnothing }$ being relatively open in $\mathrm{\Gamma }$, ${\mathrm{\Delta }}_{\mathrm{\Gamma }}$ denotes the Laplace–Beltrami operator on $\mathrm{\Gamma }$, $\nu$ is the outward normal to $\mathrm{\Omega }$, 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.