Mathematical Research Letters

Volume 29 (2022)

Number 4

Decay rates for the damped wave equation with finite regularity damping

Pages: 1087 – 1140



Perry Kleinhenz (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)


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.

Received 27 January 2020

Received revised 25 June 2021

Accepted 4 October 2021

Published 23 February 2023