A remark on the logarithmic decay of the damped wave and Schrödinger equations on a compact Riemannian manifold

Abstract

In this paper, we consider a compact Riemannian manifold (M,g) of class C^1 \cap W^{2,\infty} and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x) . We establish the following fact: if the measure of the set \{ x \in M; \, a(x) \not = 0 \} is strictly positive, then the decay in time of the associated energy is at least logarithmic.