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.

29 pages. We corrected an error in the formulation of Theorem 4(iii). This error has no influence on other parts of the paper. We extended the paper slightly by adding a characterization of the range of A (Section 3.4) in case 0 is a spectral point

References in corpus (2)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Quantified versions of Ingham's theorem

Cited by in corpus (3)

• Optimal rates of decay for operator semigroups on Hilbert spaces
• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Resolvent estimates for the one-dimensional damped wave equation with unbounded damping