$C^m$-theory of damped wave equations with stabilisation

TL;DR

This paper extends energy decay estimates for damped wave equations to more general time-dependent dissipation with oscillations, using stabilisation conditions on the damping coefficient.

Contribution

It introduces a broader class of damping functions for wave equations and applies stabilisation techniques to establish decay estimates in more complex scenarios.

Findings

Extended decay estimates to oscillatory damping functions

Established stabilisation conditions for time-dependent coefficients

Broadened applicability of energy decay results in wave equations

Abstract

The aim of this note is to extend the energy decay estimates from [J. Wirth, J. Differential Equations 222 (2006) 487--514] to a broader class of time-dependent dissipation including very fast oscillations. This is achieved using stabilisation conditions on the coefficient in the spirit of [F. Hirosawa, Math. Ann. 339/4 (2007) 819--839].