Decay of solutions of the wave equation outside rough surfaces

In the first part of this paper, we prove the decay of local energy for the solutions of the wave equation in an exterior domain outside a two-dimensional rough surface in R3 which satisfies an additional geometric condition, “weak star-shaped” condition WSS, implying the absence of trapped rays. Moreover, if the stronger “star-shaped” condition SS of Morawetz is added, the rate of decay can be bounded by 1/t2. We also remark that the result of Ralston remains valid in the “rough surface” case: the existence of trapped rays implies an arbitrarily slow decay of the energy. If we restrict the analysis to a compact perturbation of a plane, we show that the local energy decays. Moreover, if a geometric condition is added (“star-shaped” condition SS, or “nontrapping” condition NT), we find that the decay is exponential.