On the decay rate for the wave equation with viscoelastic boundary damping

TL;DR

This paper investigates the decay rates of energy in the wave equation with boundary memory damping, establishing resolvent estimates and demonstrating sharpness of results for specific geometries.

Contribution

It introduces a method to estimate resolvent bounds for the wave equation with viscoelastic boundary damping, linking decay rates to boundary impedance properties.

Findings

Established energy decay rates using resolvent estimates.

Reduced the problem to stationary resolvent estimation.

Proved sharpness of estimates for interval and disk geometries.

Abstract

We consider the wave equation with a boundary condition of memory type. Under natural conditions on the acoustic impedance $\hat{k}$ of the boundary one can define a corresponding semigroup of contractions (Desch, Fasangova, Milota, Probst 2010). With the help of Tauberian theorems we establish energy decay rates via resolvent estimates on the generator $-\mathcal{A}$ of the semigroup. We reduce the problem of estimating the resolvent of $-\mathcal{A}$ to the problem of estimating the resolvent of the corresponding stationary problem. Under not too strict additional assumptions on $\hat{k}$ we establish an upper bound on the resolvent. For the wave equation on the interval or the disk we prove our estimates to be sharp.