Lifespan of classical solutions to one dimensional fully nonlinear wave equations with time-dependent damping

This paper deals with the lower bound estimate for the maximal time T˜(ε) of the classical solutions to one dimensional fully nonlinear wave equations □u+b(t)ut=F(u,Du,DxDu) with Cauchy data of small size ε, where the coefficient b(t)=μ(1+t)−α is “effective” with μ>0 and α∈(0,1). If F is sufficie...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2022-12, Vol.516 (1), p.126470, Article 126470
Main Authors: Xiao, Changwang, Guo, Fei
Format: Article
Language:English
Subjects:
Global iteration method
Lifespan
Lower bound estimate
Nonlinear wave equation
Time-dependent damping
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Summary:This paper deals with the lower bound estimate for the maximal time T˜(ε) of the classical solutions to one dimensional fully nonlinear wave equations □u+b(t)ut=F(u,Du,DxDu) with Cauchy data of small size ε, where the coefficient b(t)=μ(1+t)−α is “effective” with μ>0 and α∈(0,1). If F is sufficiently smooth and vanishes at order p+1 in a neighborhood of zero, it is proved thatT˜(ε){≥Cε−p1−K,K1, where K=(p−1)(1+α)2 and C is a positive constant independent of ε.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126470