American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[1]

Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279
[2]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233
[3]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[4]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235
[5]

Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711
[6]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071
[7]

Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043
[8]

Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy. Electronic Research Archive, 2020, 28 (2) : 671-689. doi: 10.3934/era.2020035
[9]

Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315
[10]

Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889
[11]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[12]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076
[13]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
[14]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359
[15]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251
[16]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[17]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[18]

Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085
[19]

Qinghua Luo. Damped Klein-Gordon equation with variable diffusion coefficient. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3959-3974. doi: 10.3934/cpaa.2021139
[20]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (215)
  • HTML views (751)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy