Damped wave equations with dynamic boundary conditions

TL;DR

This paper studies linear damped wave equations with dynamic boundary conditions, analyzing well-posedness, stability, and analyticity of solutions, with applications to domains and networks.

Contribution

It introduces a comprehensive analysis of damped wave equations with dynamic boundary conditions, including well-posedness and semigroup analyticity characterization.

Findings

Solutions are well-posed under certain conditions.

Stability and boundedness depend on damping terms.

Analyticity of associated semigroups is characterized.

Abstract

We discuss several classes of linear second order initial-boundary value problems, where damping terms appear in the main wave equation as well as in the dynamic boundary condition. We investigate their well-posedness and describe some qualitative properties of their solutions, including boundedness, stability, or almost periodicity. In particular, we are able to characterize the analyticity of certain $C_0$-semigroups associated to such problems. Applications to several problems on domains and networks are shown.