  In this paper we consider a compact Riemannian manifold (M, g) of class C 1
$\cap$ W 2,$\infty$ and the damped wave or Schr\"odinger 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.
