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A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential
Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50,20133 Milano, Italy |
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.
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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