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Decay rates for the damped wave equation with finite regularity damping · arXivDesk

1910.06372Oct 14, 201944 pages. There are substantial updates from the previous version. The decay rate now depends on the regularity of the damping and the decay rate for smooth damping is improved

Decay rates for the damped wave equation with finite regularity damping

Perry Kleinhenz

Analysis of PDEs Mathematical Physics

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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 $1/t^{4/5-δ}$ . The proof uses a positive commutator argument and relies on a pseudodifferential calculus for low regularity symbols.

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The paper addresses: Decay rates for the damped wave equation with finite regularity damping.
Decay rates for the energy of solutions of the damped wave equation on the torus are studied.

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Author comment: 44 pages. There are substantial updates from the previous version. The decay rate now depends on the regularity of the damping and the decay rate for smooth damping is improved

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