Damped Wave Equations on Compact Hyperbolic Surfaces

Abstract

We prove exponential decay of energy for solutions of the damped wave equation on compact hyperbolic surfaces with regular initial data as long as the damping is nontrivial. The proof is based on a similar strategy as in Dyatlov and Jin (Acta Math 220:297–339, DyJi18) and in particular, uses the fractal uncertainty principle proved in Bourgain and Dyatlov (Ann Math (2) 187:825–867, BoDy18).