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 $$\{\begin{array}{llccc}{u}_{tt}-\Delta u+P(x,u_t)=f(x,u)& \text{in (0,∞)×Ω,}u=0& \text{on (0,∞)×Γ0,}{u}_{tt}+{\partial }_{\nu }u-{\Delta }_{\Gamma }u+Q(x,u_t)=g(x,u)& \text{on (0,∞)×Γ1,}u(0,x)=u_0(x),u_t(0,x)=u_1(x)& \text{in Ωˉ,}\end{array}$$ 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 measurable partition of $\Gamma$, ${\Delta }_{\Gamma }$ 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, 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.

This version essentially extends previous one, since an entire new section on global existence, uniqueness and Hadamrd well-posedness is added

References in corpus (4)

• On the the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and source
• Generalized Wentzell boundary conditions and quantum field theory
• Blow-up for the wave equation with nonlinear source and boundary damping terms
• Lagrangian Variational Framework for Boundary Value Problems

Cited by in corpus (7)

• Blow--up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources
• Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems
• Blow-up problems for a parabolic equation coupled with superlinear source and local linear boundary dissipation
• Three evolution problems modelling the interaction between acoustic waves and non-locally reacting surfaces
• Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition
• Acoustic waves interacting with non--locally reacting surfaces in a Lagrangian framework
• Global well-posedness for nonlinear wave equations with supercritical source and damping terms