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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 $$ \begin{cases} u_{tt}-Δu+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\timesΩ$,}\\ u=0 &\text{on $(0,\infty)\times Γ_0$,}\\ u_{tt}+\partial_νu-Δ_Γu+Q(x,u_t)=g(x,u)\qquad &\text{on $(0,\infty)\times Γ_1$,}\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & \text{in $\overlineΩ$,} \end{cases}$$ where $Ω$ is a bounded open $C^1$ subset of $\mathbb{R}^N$, $N\ge 2$, $Γ=\partialΩ$, $(Γ_0,Γ_1)$ is a partition of $Γ$, $Γ_1\not=\emptyset$ being relatively open in $Γ$, $Δ_Γ$ denotes the Laplace--Beltrami operator on $Γ$, $ν$ is the outward normal to $Ω$, 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.