Decay for the Kelvin–Voigt damped wave equation: Piecewise smooth damping

Abstract

We study the energy decay rate of the Kelvin–Voigt damped wave equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric assumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in Liu and Rao [Z. Angew. Math. Phys. 56 (2005), no. 4, 630–644]. This optimal decay rate is saturated by high energy quasi-modes localized on geometric optics rays which hit the interface along non-orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.