Abstract wave equations with acoustic boundary conditions

TL;DR

This paper develops an abstract framework for wave equations with time-dependent acoustic boundary conditions, proving well-posedness and spectral properties, and confirming a previous conjecture in the field.

Contribution

It introduces a novel abstract setting for wave equations with dynamic boundary conditions, establishing well-posedness and spectral theory, and resolving a prior conjecture.

Findings

Proved well-posedness of wave equations with acoustic boundary conditions.

Developed a spectral theory applicable to these equations.

Confirmed a conjecture from previous research.

Abstract

We define an abstract setting to treat wave equations equipped with time-dependent acoustic boundary conditions on bounded domains of ${\bf R}^n$. We prove a well-posedness result and develop a spectral theory which also allows to prove a conjecture proposed in (Gal-Goldstein-Goldstein, J. Evol. Equations 3 (2004), 623-636). Concrete problems are also discussed.