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November  2018, 23(9): 3573-3594. doi: 10.3934/dcdsb.2017215

A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential

Stefano Pasquali ,

Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50,20133 Milano, Italy

* Corresponding author: Stefano Pasquali

Received  May 2017 Revised  June 2017 Published  November 2018 Early access  September 2017

We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small analytic norm remain small for exponentially long times. The result is uniform with respect to $c ≥ 1$, which however has to belong to a set of large measure.

Keywords: Nekhoroshev theorem, nonlinear Klein-Gordon equation, Hamiltonian systems, normal forms, long time behaviour.
Mathematics Subject Classification: Primary: 35Q40, 37K45; Secondary: 37K55.
Citation: Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215
References:
[1]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Mathematische Zeitschrift, 230 (1999), 345-387.  doi: 10.1007/PL00004696.

[2]

D. Bambusi, On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity, 12 (1999), 823-850.  doi: 10.1088/0951-7715/12/4/305.

[3]

D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Communications in Mathematical Physics, 234 (2003), 253-285. 

[4]

D. Bambusi, J.-M. Delort, B. Grébert and J. Szeftel, Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Communications on Pure and Applied Mathematics, 60 (2007), 1665-1690.  doi: 10.1002/cpa.20181.

[5]

D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Mathematical Journal, 135 (2006), 507-567.  doi: 10.1215/S0012-7094-06-13534-2.

[6]

D. Bambusi and N. N. Nekhoroshev, A property of exponential stability in nonlinear wave equations near the fundamental linear mode, Physica D: Nonlinear Phenomena, 122 (1998), 73-104.  doi: 10.1016/S0167-2789(98)00169-9.

[7]

D. Bambusi and N. N. Nekhoroshev, Long time stability in perturbations of completely resonant PDE's, Acta Applicandae Mathematica, 70 (2002), 1-22.  doi: 10.1023/A:1013943111479.

[8]

G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mechanics and Dynamical Astronomy, 37 (1985), 1-25. 

[9]

J.-M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus, Journal d'Analyse Mathématique, 107 (2009), 161-194.  doi: 10.1007/s11854-009-0007-2.

[10]

J.-M. Delort and R. Imekraz, Long time existence for the semi-linear Klein-Gordon equation on a compact boundaryless Riemannian manifold, Communications in Partial Differential Equations, 42 (2017), 388-416.  doi: 10.1080/03605302.2017.1278772.

[11]

J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, International Mathematics Research Notices, 37 (2004), 1897-1966.  doi: 10.1155/S1073792804133321.

[12]

D. Fang, Z. Han and Q. Zhang, Almost global existence for the semi-linear Klein-Gordon equation on the circle, Journal of Differential Equations, 262 (2017), 4610-4634.  doi: 10.1016/j.jde.2016.12.013.

[13]

D. Fang and Q. Zhang, Long-time existence for semi-linear Klein-Gordon equations on tori, Journal of Differential Equations, 249 (2010), 151-179.  doi: 10.1016/j.jde.2010.03.025.

[14]

E. Faou and B. Grébert, A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Analysis & PDE, 6 (2013), 1243-1262.  doi: 10.2140/apde.2013.6.1243.

[15]

E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part Ⅱ. Abstract splitting, Numerische Mathematik, 114 (2010), 459-490.  doi: 10.1007/s00211-009-0257-z.

[16]

P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Mathematical Surveys, 47 (1992), 57-133.  doi: 10.1070/RM1992v047n06ABEH000965.

[17]

K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations European Mathematical Society, 2011. doi: 10.4171/095.

[18]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5-66. 

[19]

S. Pasquali, Long time behaviour of the nonlinear Klein-Gordon equation in the nonrelativistic limit, Ⅱ, preprint, arXiv: 1703.01618.

[20]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Mathematische Zeitschrift, 213 (1993), 187-216.  doi: 10.1007/BF03025718.

[21]

J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, 12 (1999), 1587-1600.  doi: 10.1088/0951-7715/12/6/310.

[22]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regular and Chaotic Dynamics, 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.

show all references

References:
[1]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Mathematische Zeitschrift, 230 (1999), 345-387.  doi: 10.1007/PL00004696.

[2]

D. Bambusi, On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity, 12 (1999), 823-850.  doi: 10.1088/0951-7715/12/4/305.

[3]

D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Communications in Mathematical Physics, 234 (2003), 253-285. 

[4]

D. Bambusi, J.-M. Delort, B. Grébert and J. Szeftel, Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Communications on Pure and Applied Mathematics, 60 (2007), 1665-1690.  doi: 10.1002/cpa.20181.

[5]

D. Bambusi and B. Grébert, Birkhoff normal form for partial differential equations with tame modulus, Duke Mathematical Journal, 135 (2006), 507-567.  doi: 10.1215/S0012-7094-06-13534-2.

[6]

D. Bambusi and N. N. Nekhoroshev, A property of exponential stability in nonlinear wave equations near the fundamental linear mode, Physica D: Nonlinear Phenomena, 122 (1998), 73-104.  doi: 10.1016/S0167-2789(98)00169-9.

[7]

D. Bambusi and N. N. Nekhoroshev, Long time stability in perturbations of completely resonant PDE's, Acta Applicandae Mathematica, 70 (2002), 1-22.  doi: 10.1023/A:1013943111479.

[8]

G. Benettin, L. Galgani and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mechanics and Dynamical Astronomy, 37 (1985), 1-25. 

[9]

J.-M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus, Journal d'Analyse Mathématique, 107 (2009), 161-194.  doi: 10.1007/s11854-009-0007-2.

[10]

J.-M. Delort and R. Imekraz, Long time existence for the semi-linear Klein-Gordon equation on a compact boundaryless Riemannian manifold, Communications in Partial Differential Equations, 42 (2017), 388-416.  doi: 10.1080/03605302.2017.1278772.

[11]

J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, International Mathematics Research Notices, 37 (2004), 1897-1966.  doi: 10.1155/S1073792804133321.

[12]

D. Fang, Z. Han and Q. Zhang, Almost global existence for the semi-linear Klein-Gordon equation on the circle, Journal of Differential Equations, 262 (2017), 4610-4634.  doi: 10.1016/j.jde.2016.12.013.

[13]

D. Fang and Q. Zhang, Long-time existence for semi-linear Klein-Gordon equations on tori, Journal of Differential Equations, 249 (2010), 151-179.  doi: 10.1016/j.jde.2010.03.025.

[14]

E. Faou and B. Grébert, A Nekhoroshev-type theorem for the nonlinear Schrödinger equation on the torus, Analysis & PDE, 6 (2013), 1243-1262.  doi: 10.2140/apde.2013.6.1243.

[15]

E. Faou, B. Grébert and E. Paturel, Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part Ⅱ. Abstract splitting, Numerische Mathematik, 114 (2010), 459-490.  doi: 10.1007/s00211-009-0257-z.

[16]

P. Lochak, Canonical perturbation theory via simultaneous approximation, Russian Mathematical Surveys, 47 (1992), 57-133.  doi: 10.1070/RM1992v047n06ABEH000965.

[17]

K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations European Mathematical Society, 2011. doi: 10.4171/095.

[18]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5-66. 

[19]

S. Pasquali, Long time behaviour of the nonlinear Klein-Gordon equation in the nonrelativistic limit, Ⅱ, preprint, arXiv: 1703.01618.

[20]

J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Mathematische Zeitschrift, 213 (1993), 187-216.  doi: 10.1007/BF03025718.

[21]

J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, 12 (1999), 1587-1600.  doi: 10.1088/0951-7715/12/6/310.

[22]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regular and Chaotic Dynamics, 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.

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