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$L^{2}$ -blowup estimates of the wave equation and its application to local energy decay

Ryo Ikehata

Department of Mathematics, Division of Educational Sciences, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima 739-8524, Japan

E-mail Address: ikehatar@hiroshima-u.ac.jp

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https://doi.org/10.1142/S021989162350008XCited by:9 (Source: Crossref)

Abstract

We consider the Cauchy problems in $R^{n}$ for the wave equation with a weighted $L^{1}$ -initial data. We derive sharp infinite time blowup estimates of the $L^{2}$ -norm of solutions in the case of $n = 1$ and $n = 2$ . Then, we apply it to the local energy decay estimates for $n = 2$ , which is not studied so completely when the $0$ th moment of the initial velocity does not vanish. The idea to derive them is strongly inspired from a technique used in [R. Ikehata, Asymptotic profiles for wave equations with strong damping, J. Differ. Equ. 257 (2014) 2159–2177; R. Ikehata and M. Onodera, Remarks on large time behavior of the $L^{2}$ -norm of solutions to strongly damped wave equations, Differ. Integral Equ. 30 (2017) 505–520].

Communicated by S. Kawashima

Keywords:

AMSC: 35L05, 35B40, 35L20

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Vol. 20, No. 01

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History

Received 22 April 2022

Accepted 13 November 2022

Published: 19 May 2023

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Open Access since May 2023. This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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$L^{2}$ -blowup estimates of the wave equation and its application to local energy decay