Decays rates for Kelvin-Voigt damped wave equations II: the geometric control condition

TL;DR

This paper investigates decay rates for Kelvin-Voigt damped wave equations under geometric control conditions, demonstrating exponential decay when the damping coefficient is sufficiently smooth and vanishes appropriately.

Contribution

It establishes exponential decay results for Kelvin-Voigt damped wave equations with smooth damping coefficients under geometric control conditions, extending previous results with weaker assumptions.

Findings

Exponential decay achieved under geometric control condition

Smooth damping coefficient with specific vanishing properties suffices

Extends prior results to less restrictive damping assumptions

Abstract

We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. We prove that when the damping coefficient is sufficiently smooth ($C^1$ vanishing nicely) we show that exponential decay follows from geometric control conditions (see~\cite{BuCh, Te12} for similar results under stronger assumptions on the damping function).