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 $\{x\in M;a(x)\ne 0\}$ is strictly positive, then the decay in time of the associated energy is at least logarithmic.