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Elliptic problems with rough boundary data in generalized Sobolev spaces
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 |
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 $. The proof relies on the method of normal forms and induction. The structure of the nonlinearity is good enough that proceeds normal forms up to any order.
References:
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On the well-posedness of the Kirchhoff string, T. Am. Math. Soc., 348 (1996), 305-330.
doi: 10.1090/S0002-9947-96-01532-2. |
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Sur une classe d'$\acute{e}$quations fonctionnelles aux d$\acute{e}$riv$\acute{e}$es partielles, Izv. Akad. Nauk SSSR Ser. Mat., 4 (1940), 17-26.
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On the existence time for the Kirchhoff equation with periodic boundary conditions, Nonlinearity, 33 (2020), 196-223.
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R. W. Dickey,
Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc., 23 (1969), 459-468.
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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. |
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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. |
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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. |
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G. Kirchhoff, Vorlesungen $\ddot{u}$ber mathematische Physik: Mechanik, ch. 29, Teubner, Leipzig, 1876. |
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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.
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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. |
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S. Spagnolo,
The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano, 62 (1992), 17-51.
doi: 10.1007/bf02925435. |
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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. |
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. |
| [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. |
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