A remark on the logarithmic decay of the damped wave and Schr\"odinger equations on a compact Riemannian manifold

TL;DR

This paper proves that on a compact Riemannian manifold, if the damping function vanishes on a set of positive measure, then the energy decay of damped wave and Schrödinger equations is at least logarithmic over time.

Contribution

It establishes a lower bound of logarithmic decay for energy in damped wave and Schrödinger equations under specific geometric conditions.

Findings

Energy decay is at least logarithmic when the damping set has positive measure.

The result applies to manifolds with C1∩W2,∞ regularity.

The decay rate depends on the measure of the undamped set.

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 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.