Affiliations: Department of Mathematics, Texas A & M University, College Station, TX 77843, USA | Department of Mathematics, Sichuan University, Chengdu, Sichuan, People's Republic of China | Department of Mathematics, National Central University, Chung-Li, Taiwan, ROC
Note: [] Supported in part by AFOSR Grants 87-0334 and 91-0097.
Note: [] Supported by the National Foundation of China.
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.
