Decay rates for the damped wave equation with finite regularity damping

Abstract: Decay rates for the energy of solutions of the damped wave equation on the torus are studied. In particular, damping invariant in one direction and equal to a sum of squares of nonnegative functions with a particular number of derivatives of regularity is considered. For such damping energy decays at rate $1/t^{2/3}$. If additional regularity is assumed the decay rate improves. When such a damping is smooth the energy decays at $1/t^{4/5-\delta}$. The proof uses a positive commutator argument and relies on a pseudodifferential calculus for low regularity symbols.