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 $$\{\begin{array}{ll}{u}_{tt}-\Delta u+P(x,u_t)=f(x,u)& \text{in (0,∞)×Ω,}\\ u=0& \text{on (0,∞)×Γ0,}\\ u_{tt}+{\partial }_{\nu }u-{\Delta }_{\Gamma }u+Q(x,u_t)=g(x,u)& \text{on (0,∞)×Γ1,}\\ u(0,x)=u_0(x),u_t(0,x)=u_1(x)& \text{in Ω,}\end{array}$$ 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 $\Gamma$, ${\Gamma }_1\ne \varnothing$ 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.

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