Decay for the Kelvin-Voigt damped wave equation: Piecewise smooth damping

TL;DR

This paper investigates the energy decay rates of the Kelvin-Voigt damped wave equation with piecewise smooth damping in multi-dimensional domains, revealing optimal polynomial decay rates influenced by geometric conditions.

Contribution

It establishes the optimal polynomial decay rate for multi-dimensional Kelvin-Voigt damped wave equations with piecewise smooth damping, extending previous one-dimensional results.

Findings

Optimal polynomial decay rate derived for multi-dimensional case

Decay rate influenced by geometric support of damping

High energy quasi-modes localized on geometric optics rays

Abstract

We study the energy decay rate of the Kelvin-Voigt damped wave equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric assumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in \cite{LR05}. This optimal decay rate is saturated by high energy quasi-modes localised on geometric optics rays which hit the interface along non orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.