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arXiv · 1701.01013

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

Reinhard Stahn
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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.

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Reinhard Stahn. 2017-09-05. On the decay rate for the wave equation with viscoelastic boundary damping. https://doi.org/10.1016/j.jde.2018.04.048

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