Stabilization of Damped Waves on Spheres and Zoll Surfaces of Revolution

TL;DR

This paper demonstrates strong stabilization of wave equations on sphere-like manifolds using rough damping regions that do not satisfy traditional geometric control conditions, extending known results from spheres to Zoll surfaces of revolution.

Contribution

It extends Lebeau's stabilization result from spheres to Zoll surfaces of revolution, showing that partial damping regions can still achieve strong stabilization despite geometric control condition failures.

Findings

Strong stabilization achieved on S^d with upper hemisphere damping.

Extension of stabilization results to Zoll surfaces of revolution.

Damping regions do not need to contain all geodesics for stabilization.

Abstract

We study the strong stabilization of wave equations on some sphere-like manifolds, with rough damping terms which do not satisfy the geometric control condition posed by Rauch-Taylor and Bardos-Lebeau-Rauch. We begin with an unpublished result of G. Lebeau, which states that on S^d , the indicator function of the upper hemisphere strongly stabilizes the damped wave equation, even though the equators, which are geodesics contained in the boundary of the upper hemisphere, do not enter the damping region. Then we extend this result on dimension 2, to Zoll surfaces of revolution, whose geometry is similar to that of S^2 . In particular, geometric objects such as the equator, and the hemi-surfaces are well defined. Our result states that the indicator function of the upper hemi-surface strongly stabilizes the damped wave equation, even though the equator, as a geodesic, does not enter the upper hemi-surface either.