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Decay estimate for subcritical semilinear damped wave equations with slowly decreasing data

Part of: Partial differential equations

Published online by Cambridge University Press:  02 January 2026

Kazumasa Fujiwara Open the ORCID record for Kazumasa Fujiwara [Opens in a new window]  and
Vladimir Georgiev Open the ORCID record for Vladimir Georgiev [Opens in a new window]
Kazumasa Fujiwara*
Affiliation:
Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-cho, Otsu, Shiga, 520-2194, Japan (fujiwara.kazumasa@math.ryukoku.ac.jp)
Vladimir Georgiev
Affiliation:
Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, I - 56127 Pisa, Italy Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, Sofia, 1113, Bulgaria (vladimir.simeonov.gueorguiev@unipi.it)
*
*Corresponding author.
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Abstract

We study the decay properties of non-negative solutions to the one-dimensional defocusing damped wave equation in the Fujita subcritical case under a specific initial condition. Specifically, we assume that the initial data are positive, satisfy a condition ensuring the positiveness of solutions, and exhibit polynomial decay at infinity. To show the decay properties of the solution, we construct suitable supersolutions composed of an explicit function satisfying an ordinary differential inequality and the solution of the linear damped wave equation. Our estimates correspond to the optimal ones inferred from the analysis of the heat equation.

Keywords

damped wave equationsdecay estimateFujita critical exponentpower-type nonlinearityFujita subcritical case

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35B33: Critical exponents 35B51: Comparison principles

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 21
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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Footnotes

‡

This article originally published with an error. The error has since been corrected and a corrigendum published.

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