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Proceedings of the American Mathematical Society

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Sharp polynomial decay rates for the damped wave equation with Hölder-like damping


Authors: Kiril Datchev and Perry Kleinhenz
Journal: Proc. Amer. Math. Soc. 148 (2020), 3417-3425
MSC (2010): Primary 35L05, 47A10
DOI: https://doi.org/10.1090/proc/15018
Published electronically: May 8, 2020
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Abstract: We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $ x^{\beta }$ near the boundary of the support and show decay at rate $ 1/t^{\frac {\beta +2}{\beta +3}}$. In the case where $ W$ vanishes exactly like $ x^{\beta }$ this result is optimal by [Comm. Math. Phys. 369 (2019), pp. 1187-1205]. The proof uses a version of the Morawetz multiplier method.


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Additional Information

Kiril Datchev
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: kdatchev@purdue.edu

Perry Kleinhenz
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: pbk@math.northwestern.edu

DOI: https://doi.org/10.1090/proc/15018
Received by editor(s): September 6, 2019
Received by editor(s) in revised form: December 18, 2019
Published electronically: May 8, 2020
Additional Notes: The first author was partially supported by NSF Grant DMS-1708511.
The second author was partially supported by the National Science Foundation grant RTG: Analysis on Manifolds at Northwestern University.
Communicated by: Ariel Barton
Article copyright: © Copyright 2020 American Mathematical Society