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

We consider the wave equation with a boundary condition of memory type. Under natural conditions on the acoustic impedance of the boundary a corresponding semigroup of contractions is known to exist. With the help of quantified Tauberian theorems we establish energy decay rates via resolvent estimates on the generator of the semigroup. Using a variational approach, we reduce resolvent estimates to estimates for a sesquilinear form induced by an operator characteristic function arising form the matrix representation of the generator. Under not too strict additional assumptions on the acoustic impedance we establish an upper bound on the resolvent. For the wave equation on the interval or the disc we prove our estimates to be sharp. (C) 2018 Elsevier Inc. All rights reserved.