On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources

Abstract

The aim of the paper is to study the problem $$ \begin{cases} u_{tt}-Δu+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\timesΩ$,} u=0 &\text{on $(0,\infty)\times Γ_0$,} u_{tt}+\partial_νu-Δ_Γu+Q(x,u_t)=g(x,u)\qquad &\text{on $(0,\infty)\times Γ_1$,} u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) & \text{in $\barΩ$,} \end{cases}$$ where $Ω$ is a bounded open $C^1$ subset of $\mathbb{R}^N$, $N\ge 2$, $Γ=\partialΩ$, $(Γ_0,Γ_1)$ is a measurable partition of $Γ$, $Δ_Γ$ 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, or sink, terms. In the paper we establish local and existence, uniqueness and Hadamard well--posedness results when source terms can be supercritical or super-supercritical.