February  2021, 20(2): 737-754. doi: 10.3934/cpaa.2020287

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

1. 

Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China

2. 

Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

* Corresponding author

Received  May 2020 Revised  October 2020 Published  December 2020

Fund Project: 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

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.

Citation: Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Y. FangZ. 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.  Google Scholar

[8]

G. Kirchhoff, Vorlesungen $\ddot{u}$ber mathematische Physik: Mechanik, ch. 29, Teubner, Leipzig, 1876. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

S. Spagnolo, The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano, 62 (1992), 17-51.  doi: 10.1007/bf02925435.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. Y. FangZ. 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.  Google Scholar

[8]

G. Kirchhoff, Vorlesungen $\ddot{u}$ber mathematische Physik: Mechanik, ch. 29, Teubner, Leipzig, 1876. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[11]

S. Spagnolo, The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano, 62 (1992), 17-51.  doi: 10.1007/bf02925435.  Google Scholar

[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.  Google Scholar

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