Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

TL;DR

This paper investigates the long-term behavior of solutions to a multi-dimensional damped wave equation with dynamic boundary conditions, establishing conditions for stability and blow-up phenomena.

Contribution

It provides new results on global existence, asymptotic stability, and blow-up for solutions of a wave equation with Kelvin-Voigt damping and dynamic boundary conditions.

Findings

Proved global existence and stability for solutions starting in a stable set.

Established blow-up results for solutions with initial data in an unstable set.

Analyzed the influence of dynamic boundary conditions on solution behavior.

Abstract

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.