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

In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $\cap$ W 2,$\mathrm{\infty }$ 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) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.