“More boundary energy dissipation” does not necessarily imply larger asymptotic decay rates

Abstract

For the wave equation with an impedance boundary condition in a bounded domain, it is known that its energy will decay uniformly exponentially provided that the domain has, e.g., a star-shaped geometry (Chen, 1979; Lagnese, 1983). So it appears natural at first sight that the energy decay rates should increase when there is additional energy dissipation on the boundary contributed by higher-order tangential derivatives. In this paper, we study the evolution of the wave equation with such higher-order tangential dissipative boundary conditions. By using a geometrical optics expansion and a wave method argument, we show by an example on a 2-dimensional disk that at high frequencies the highest-order derivative in the boundary condition is dominant, and there is no uniform exponential decay for such modes. This constitutes a counterexample to the seemingly natural “comparison theorem” for energy decay properties.