Stabilization rates for the damped wave equation with H\"older-regular damping

TL;DR

This paper investigates how the polynomial vanishing of damping coefficients affects the decay rates of solutions to the damped wave equation, providing bounds that relate damping regularity to energy decay.

Contribution

It establishes explicit upper and lower bounds on decay rates for the damped wave equation with polynomial damping vanishing, generalizing previous results with indicator damping.

Findings

Decay rate cannot be faster than 1/t^{(eta+2)/(eta+3)}

Decay rate is at least 1/t^{(eta+2)/(eta+4)}

Results relate damping vanishing rate to energy decay rate

Abstract

We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. We consider damping coefficients which are $0$ on a strip and vanish like polynomials, $x^{\beta}$. We prove that the semigroup cannot be stable at rate faster than $1/t^{(\beta+2)/(\beta+3)}$ by producing quasimodes of the associated stationary damped wave equation. We also prove that the semigroup is stable at rate at least as fast as $1/t^{(\beta+2)/(\beta+4)}$. These two results establish an explicit relation between the rate of vanishing of the damping and rate of decay of solutions. Our result partially generalizes a decay result of Nonnemacher in which the damping is an indicator function on a strip.