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

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.