Stabilization of damped waves on spheres and Zoll surfaces of revolution

Zhu, Hui

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1759-1788.

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 [J. Rauch and M. Taylor, Commun. Pure Appl. Math. 28 (1975) 501–523] and Bardos−Lebeau−Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optimiz. 30 (1992) 1024–1065]. We begin with an unpublished result of G. Lebeau, which states that on ${𝕊}^d$, the indicator function of the upper hemisphere strongly stabilizes the damped wave equation, even though the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.

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