November 2018, 38(11): 5765-5780. doi: 10.3934/dcds.2018251

On dispersion decay for 3D Klein-Gordon equation

1. 

Faculty of Mathematics, Vienna University, Austria

2. 

Institute for Information Transmission Problems RAS, Moscow, Russia

* Corresponding author: Elena Kopylova

Received  January 2018 Revised  April 2018 Published  August 2018

Fund Project: The author is supported by by the Austrian Science Fund (FWF) under Grant No. P27492-N25 and RFBR grants

We improve previous results on dispersion decay for 3D KleinGordon equation with generic potential. We develop a novel approach, which allows us to establish the decay in more strong norms and to weaken assumptions on the potential.

Citation: Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251
References:
[1]

S. Agmon, Spectral properties of Schrödinger operator and scattering theory, Ann. Scuola Norm. Sup. Pisa, Ser. Ⅳ, 2 (1975), 151-218.

[2]

P. D'Ancona, On Large potential perturbation of the Schrödinger, wave and Klein-Gordon equations, arXiv: 1706.04840v2.

[3]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112. doi: 10.1080/03605300701743749.

[4]

N. Boussaid and S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), 1001-1056. doi: 10.1080/03605302.2012.665973.

[5]

S. Cuccagna and T. Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys., 284 (2008), 51-77. doi: 10.1007/s00220-008-0605-3.

[6]

M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. doi: 10.1007/s00220-004-1140-5.

[7]

V. ImaikinA. Komech and B. Vainberg, On scattering of solitons for the Klein-Gordon equation coupled to a particle, Comm. Math. Phys., 268 (2006), 321-367. doi: 10.1007/s00220-006-0088-z.

[8]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.

[9]

A. Komech and E. Kopylova, Weighted energy decay for 3D Klein-Gordon equation, J. of Diff. Eqns., 248 (2010), 501-520. doi: 10.1016/j.jde.2009.06.011.

[10]

A. Komech and E. Kopylova, Dispersion Decay and Scattering Theory, John Willey & Sons, Hoboken, New Jersey, 2012. doi: 10.1002/9781118382868.

[11]

E. Kopylova and A. Komech, On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation, Comm. Math. Phys., 302 (2011), 225-252. doi: 10.1007/s00220-010-1184-7.

[12]

E. Kopylova and A. Komech, On asymptotic stability of kink for relativistic Ginzburg-Landau equation, Arch. Ration. Mech. Anal., 202 (2011), 213-245. doi: 10.1007/s00205-011-0415-1.

[13]

H. Kubo and S. Lucente, Note on weighted Strichartz estimates for Klein-Gordon equations with potential, Tsukuba J. Math., 31 (2007), 143-173. doi: 10.21099/tkbjm/1496165119.

[14]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Series, 43 Princeton University Press, Princeton, NJ, 1993.

[15]

B. R. Vainberg, Behavior for large time of solutions of the Klein-Gordon equation, Trans. Mosc. Math. Soc., 30 (1976), 139-158.

show all references

References:
[1]

S. Agmon, Spectral properties of Schrödinger operator and scattering theory, Ann. Scuola Norm. Sup. Pisa, Ser. Ⅳ, 2 (1975), 151-218.

[2]

P. D'Ancona, On Large potential perturbation of the Schrödinger, wave and Klein-Gordon equations, arXiv: 1706.04840v2.

[3]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112. doi: 10.1080/03605300701743749.

[4]

N. Boussaid and S. Cuccagna, On stability of standing waves of nonlinear Dirac equations, Comm. Partial Differential Equations, 37 (2012), 1001-1056. doi: 10.1080/03605302.2012.665973.

[5]

S. Cuccagna and T. Mizumachi, On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations, Comm. Math. Phys., 284 (2008), 51-77. doi: 10.1007/s00220-008-0605-3.

[6]

M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. doi: 10.1007/s00220-004-1140-5.

[7]

V. ImaikinA. Komech and B. Vainberg, On scattering of solitons for the Klein-Gordon equation coupled to a particle, Comm. Math. Phys., 268 (2006), 321-367. doi: 10.1007/s00220-006-0088-z.

[8]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke Math. J., 46 (1979), 583-611.

[9]

A. Komech and E. Kopylova, Weighted energy decay for 3D Klein-Gordon equation, J. of Diff. Eqns., 248 (2010), 501-520. doi: 10.1016/j.jde.2009.06.011.

[10]

A. Komech and E. Kopylova, Dispersion Decay and Scattering Theory, John Willey & Sons, Hoboken, New Jersey, 2012. doi: 10.1002/9781118382868.

[11]

E. Kopylova and A. Komech, On asymptotic stability of moving kink for relativistic Ginzburg-Landau equation, Comm. Math. Phys., 302 (2011), 225-252. doi: 10.1007/s00220-010-1184-7.

[12]

E. Kopylova and A. Komech, On asymptotic stability of kink for relativistic Ginzburg-Landau equation, Arch. Ration. Mech. Anal., 202 (2011), 213-245. doi: 10.1007/s00205-011-0415-1.

[13]

H. Kubo and S. Lucente, Note on weighted Strichartz estimates for Klein-Gordon equations with potential, Tsukuba J. Math., 31 (2007), 143-173. doi: 10.21099/tkbjm/1496165119.

[14]

E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Series, 43 Princeton University Press, Princeton, NJ, 1993.

[15]

B. R. Vainberg, Behavior for large time of solutions of the Klein-Gordon equation, Trans. Mosc. Math. Soc., 30 (1976), 139-158.

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