Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions

TL;DR

This paper establishes local existence and demonstrates exponential energy growth for a multi-dimensional damped semilinear wave equation with dynamic boundary conditions, using Faedo-Galerkin methods and energy analysis.

Contribution

It introduces a novel analysis of a damped semilinear wave equation with dynamic boundary conditions, proving local existence and exponential energy growth.

Findings

Proved local existence using Faedo-Galerkin approximations.

Established exponential growth of energy and solution norms.

Analyzed a wave equation related to Kelvin-Voigt damping.

Abstract

In this paper we consider a multi-dimensional damped semiliear wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. We firstly prove the local existence by using the Faedo-Galerkin approximations combined with a contraction mapping theorem. Secondly, the exponential growth of the energy and the $L^p$ norm of the solution is presented.