Remark on One Dimensional Semilinear Damped Wave Equation in a Critical Weighted L 2-space

Motohiro Sobajima; Yuta Wakasugi

doi:10.1007/978-3-030-73363-6_14

Remark on One Dimensional Semilinear Damped Wave Equation in a Critical Weighted L ²-space

Motohiro Sobajima
, Yuta Wakasugi

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

2 Citations (Scopus)

Abstract

We study the Cauchy problem of the semilinear damped wave equation in one space dimension. We show the existence of global solutions in the critical case with small initial data in weighted L²-spaces. This problem in multidimensional cases was dealt with in Sobajima (Differ Integr Equ 32:615–638, 2019) via the weighted Hardy inequality which is false in one-dimension. The crucial idea of the proof is the use of an incomplete version of Hardy inequality.

Original language	English
Title of host publication	Springer INdAM Series
Publisher	Springer-Verlag Italia s.r.l.
Pages	291-305
Number of pages	15
DOIs	https://doi.org/10.1007/978-3-030-73363-6_14
Publication status	Published - 2021

Publication series

Name	Springer INdAM Series
Volume	47
ISSN (Print)	2281-518X
ISSN (Electronic)	2281-5198

Keywords

Critical case
Global existence
One dimension
Semilinear damped wave equations

Access to Document

10.1007/978-3-030-73363-6_14

Cite this

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abstract = "We study the Cauchy problem of the semilinear damped wave equation in one space dimension. We show the existence of global solutions in the critical case with small initial data in weighted L2-spaces. This problem in multidimensional cases was dealt with in Sobajima (Differ Integr Equ 32:615–638, 2019) via the weighted Hardy inequality which is false in one-dimension. The crucial idea of the proof is the use of an incomplete version of Hardy inequality.",

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AB - We study the Cauchy problem of the semilinear damped wave equation in one space dimension. We show the existence of global solutions in the critical case with small initial data in weighted L2-spaces. This problem in multidimensional cases was dealt with in Sobajima (Differ Integr Equ 32:615–638, 2019) via the weighted Hardy inequality which is false in one-dimension. The crucial idea of the proof is the use of an incomplete version of Hardy inequality.

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KW - Global existence

KW - One dimension

KW - Semilinear damped wave equations

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