Almost global solutions of 1D nonlinear Klein-Gordon equations with small weakly decaying initial data

2023/09/28 by Fei Hou, Fei Tao, Hou, Fei +3
Mathematics · Medicine · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Soft tissue tumor case studies

paper · pdf · doi:10.48550/arxiv.2309.16213

openalex publication_date 2023/09/28 · openalex created_date 2023/10/01 · openalex updated_date 2026/07/28

Abstract

It has been known that if the initial data decay sufficiently fast at space infinity, then 1D Klein-Gordon equations with quadratic nonlinearity admit classical solutions up to time e^C/ε2 while e^C/ε2 is also the upper bound of the lifespan, where C>0 is some suitable constant and ε>0 is the size of the initial data. In this paper, we will focus on the 1D nonlinear Klein-Gordon equations with weakly decaying initial data. It is shown that if the H^s-Sobolev norm with (1+|x|)^1/2+ weight of the initial data is small, then the almost global solutions exist; if the initial H^s-Sobolev norm with (1+|x|)^1/2 weight is small, then for any M>0, the solutions exist on [0,ε^-M]. Our proof is based on the dispersive estimate with a suitable Z-norm and a delicate analysis on the phase function.

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