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# Mathematical Research Letters

## Volume 29 (2022)

### Number 4

### Decay rates for the damped wave equation with finite regularity damping

Pages: 1087 – 1140

DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n4.a8

#### Author

#### 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.

Received 27 January 2020

Received revised 25 June 2021

Accepted 4 October 2021

Published 23 February 2023