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).
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I am very grateful to Kiril Datchev, Semyon Dyatlov, and Maciej Zworski for the encouragement to work on this project and many great suggestions to the early draft of the paper. I would also like to thank Hans Christianson and Jared Wunsch for the helpful discussions about damped wave equations. The work is supported by Recruitment Program of Young Overseas Talent Plan.
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Communicated by J. Marklof
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Jin, L. Damped Wave Equations on Compact Hyperbolic Surfaces. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-019-03650-x