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

[1]	A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, T. Am. Math. Soc., 348 (1996), 305-330. doi: 10.1090/S0002-9947-96-01532-2.
[2]	S. N. Bernstein, Sur une classe d'$\acute{e}$quations fonctionnelles aux d$\acute{e}$riv$\acute{e}$es partielles, Izv. Akad. Nauk SSSR Ser. Mat., 4 (1940), 17-26.
[3]	P. Baldi and E. Haus, On the existence time for the Kirchhoff equation with periodic boundary conditions, Nonlinearity, 33 (2020), 196-223. doi: 10.1088/1361-6544/ab4c7b.
[4]	R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc., 23 (1969), 459-468. doi: 10.1090/S0002-9939-1969-0247189-8.
[5]	J. M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equaitons on the torus, J. Anal. Math., 107 (2009), 161-194. doi: 10.1007/s11854-009-0007-2.
[6]	J. M. Delort and J. Szeftel, Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Am. J. Math., 128 (2006), 1187-1218. doi: 10.1353/ajm.2006.0038.
[7]	D. Y. Fang, Z. Han and Q. D. Zhang, Almost global existence for the semi-linear Klein-Gordon equation on the circle, J. Differ. Equ., 262 (2017), 4610-4634. doi: 10.1016/j.jde.2016.12.013.
[8]	G. Kirchhoff, Vorlesungen $\ddot{u}$ber mathematische Physik: Mechanik, ch. 29, Teubner, Leipzig, 1876.
[9]	L. A. Medeiros and M. M. Miranda, Solutions for the equation of nonlinear vibrations in Sobolev spaces of fractionary order, Mat. Apl. Comput., 6 (1987), 257-276.
[10]	T. Matsuyama and M. Ruzhansky, Global well-posedness of Kirchhoff system, J. Math. Pures Appl., 100 (2013), 220-240. doi: 10.1016/j.matpur.2012.12.002.
[11]	S. Spagnolo, The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano, 62 (1992), 17-51. doi: 10.1007/bf02925435.
[12]	T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three, Math. methods Appl. Sci., 27 (2004), 1893-1916. doi: 10.1002/mma.530.