Existence and exponential stability of a damped wave equation with dynamic boundary conditions and a delay term

TL;DR

This paper investigates a multi-dimensional damped wave equation with dynamic boundary conditions and a delay term, establishing conditions for global existence and exponential stability using Lyapunov functionals.

Contribution

It introduces new stability results for wave equations with boundary delays, showing exponential stability even when delay effects dominate damping.

Findings

Global existence of solutions under specific delay conditions

Exponential stability proven via Lyapunov functional

Strong damping ensures stability despite delays

Abstract

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions related to the Kelvin-Voigt damping and a delay term acting on the boundary. If the weight of the delay term in the feedback is less than the weight of the term without delay or if it is greater under an assumption between the damping factor, and the difference of the two weights, we prove the global existence of the solutions. Under the same assumptions, the exponential stability of the system is proved using an appropriate Lyapunov functional. More precisely, we show that even when the weight of the delay is greater than the weight of the damping in the boundary conditions, the strong damping term still provides exponential stability for the system.