Operator matrices as generators of cosine operator functions

TL;DR

This paper develops an abstract framework using operator matrices to analyze wave equations with time-dependent boundary conditions, establishing well-posedness criteria and applying them to specific PDEs with dynamic and acoustic boundary conditions.

Contribution

It introduces a novel operator matrix approach to handle wave equations with time-dependent boundary conditions and characterizes their well-posedness.

Findings

Well-posedness of wave equations with time-dependent boundary conditions is characterized by perturbations of problems with homogeneous conditions.

Application to wave equations in $L^p(0,1)$ and $L^2( abla)$ demonstrates the framework's versatility.

The approach links well-posedness of time-dependent boundary problems to simpler, time-independent cases.

Abstract

We introduce an abstract setting that allows to discuss wave equations with time-dependent boundary conditions by means of operator matrices. We show that such problems are well-posed if and only if certain perturbations of the same problems with homogeneous, time-independent boundary conditions are well-posed. As applications we discuss two wave equations in $L^p(0,1)$ and in $L^2(\Omega)$ equipped with dynamical and acoustic-like boundary conditions, respectively.