Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity

Zheng Han; Daoyuan Fang

doi:10.3934/cpaa.2020287

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2021, Volume 20, Issue 2: 737-754. Doi: 10.3934/cpaa.2020287

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Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity

Zheng Han^1, , and
Daoyuan Fang^2,

1.
Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China
2.
Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

^* Corresponding author
^* Corresponding author

Received: May 2020

Revised: October 2020

Early access: December 2020

Published: February 2021

The first author is supported by NSFC grant 11671353, 11401153, Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010025. The second author is supported by NSFC grant 11671353.

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Abstract

We prove an almost global existence result for the Klein-Gordon equation with the Kirchhoff-type nonlinearity on $ \mathbb{T}^d $ with Cauchy data of small amplitude $ \epsilon $. We show a lower bound $ \epsilon^{-2N-2} $ for the existence time with any natural number $ N $. The proof relies on the method of normal forms and induction. The structure of the nonlinearity is good enough that proceeds normal forms up to any order.

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Mathematics Subject Classification: Primary:35L70, 35L15;Secondary:58J45, 70K45.

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References

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