-
Previous Article
$L^∞$-energy method for a parabolic system with convection and hysteresis effect
- CPAA Home
- This Issue
-
Next Article
On the Cauchy problem for the Zakharov-Rubenchik/ Benney-Roskes system
Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions
| 1. | Department systems innovation, Graduate school of Engineering Science, Osaka University, 1-3, Machikaneyamacho, Toyonaka, Osaka 560-8531, Japan |
| 2. | Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan |
In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: $(\Box+1)u = λ|u|^{2/3}u$, $t∈\mathbb{R}$, $x∈\mathbb{R}^{3}$, where $\Box = \partial_{t}^{2}-Δ$ is d'Alembertian. We prove that for a given asymptotic profile $u_{\mathrm{ap}}$, there exists a solution $u$ to (NLKG) which converges to $u_{\mathrm{ap}}$ as $t\to∞$. Here the asymptotic profile $u_{\mathrm{ap}}$ is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [
References:
| [1] |
J-M. Delort,
Existence globale et comportement asymptotique pour l'equation de KleinGordon quasi linéaire à données petites en dimension 1. (French), Ann. Sci. l'Ecole Norm. Sup., 34 (2001), 1-61.
|
| [2] |
J-M. Delort, D. Fang and R. Xue,
Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323.
|
| [3] |
V. Georgiev,
Decay estimates for the Klein-Gordon equation, Comm. Part. Diff. Eq., 17 (1992), 1111-1139.
|
| [4] |
V. Georgiev and S. Lecente,
Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 21-26.
|
| [5] |
V. Georgiev and B. Yardanov, Asymptotic behavior of the one dimensional Klein-Gordon equation with a cubic nonlinearity, preprint, (1996). |
| [6] |
J. Ginibre and T. Ozawa,
Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys., 151 (1993), 619-645.
|
| [7] |
R. T. Glassey,
On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200.
|
| [8] |
N. Hayashi and P. I. Naumkin,
The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.
|
| [9] |
N. Hayashi and P. I. Naumkin,
Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations, 244 (2008), 188-199.
|
| [10] |
N. Hayashi and P. I. Naumkin,
Final state problem for the cubic nonlinear Klein-Gordon equation, J. Math. Phys., 50 (2009), 103511-14 pp.
|
| [11] |
N. Hayashi and P. I. Naumkin,
Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math., 11 (2009), 771-781.
|
| [12] |
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, in: Mathématiques et Applications, 26, Springer, Berlin, 1997. |
| [13] |
S. Katayama,
A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213.
|
| [14] |
S. Katayama, T. Ozawa and H. Sunagawa,
A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math., 65 (2012), 1285-1302.
|
| [15] |
Y. Kawahara and H. Sunagawa,
Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567.
|
| [16] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
|
| [17] |
S. Klainerman,
Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641.
|
| [18] |
H. Lindblad and A. Soffer,
A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ., 1 (2005), 77-89.
|
| [19] |
H. Lindblad and A. Soffer,
A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258.
|
| [20] |
B. Marshall, W. Strauss and S. Wainger,
Lp-Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl., 59 (1980), 417-440.
|
| [21] |
S. Masaki and H. Miyazaki, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity, preprint available at arXiv: 1612.04524. |
| [22] |
S. Masaki, H. Miyazaki and K. Uriya, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimension, preprint. |
| [23] |
S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, to appear in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, preprint available at arXiv: 1602.05331. |
| [24] |
S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, to appear in Trans. AMS, preprint available at arXiv: 1612.00109. |
| [25] |
A. Matsumura,
On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189.
|
| [26] |
K. Moriyama,
Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520.
|
| [27] |
K. Moriyama, S. Tonegawa and Y. Tsutsumi,
Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math., 5 (2003), 983-996.
|
| [28] |
T. Ozawa, K. Tsutaya and Y. Tsutsumi,
Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z., 222 (1996), 341-362.
|
| [29] |
H. Pecher,
Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.
|
| [30] |
H. Pecher,
Low energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal., 63 (1985), 101-122.
|
| [31] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.
|
| [32] |
A. Shimomura and S. Tonegawa,
Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions, Differential Integral Equations, 17 (2004), 127-150.
|
| [33] |
W. A. Strauss,
Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.
|
| [34] |
H. Sunagawa,
Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400.
|
| [35] |
H. Sunagawa,
Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494.
|
| [36] |
K. Yajima,
Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
|
show all references
References:
| [1] |
J-M. Delort,
Existence globale et comportement asymptotique pour l'equation de KleinGordon quasi linéaire à données petites en dimension 1. (French), Ann. Sci. l'Ecole Norm. Sup., 34 (2001), 1-61.
|
| [2] |
J-M. Delort, D. Fang and R. Xue,
Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., 211 (2004), 288-323.
|
| [3] |
V. Georgiev,
Decay estimates for the Klein-Gordon equation, Comm. Part. Diff. Eq., 17 (1992), 1111-1139.
|
| [4] |
V. Georgiev and S. Lecente,
Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 21-26.
|
| [5] |
V. Georgiev and B. Yardanov, Asymptotic behavior of the one dimensional Klein-Gordon equation with a cubic nonlinearity, preprint, (1996). |
| [6] |
J. Ginibre and T. Ozawa,
Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension n ≥ 2, Comm. Math. Phys., 151 (1993), 619-645.
|
| [7] |
R. T. Glassey,
On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc., 182 (1973), 187-200.
|
| [8] |
N. Hayashi and P. I. Naumkin,
The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys., 59 (2008), 1002-1028.
|
| [9] |
N. Hayashi and P. I. Naumkin,
Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions, J. Differential Equations, 244 (2008), 188-199.
|
| [10] |
N. Hayashi and P. I. Naumkin,
Final state problem for the cubic nonlinear Klein-Gordon equation, J. Math. Phys., 50 (2009), 103511-14 pp.
|
| [11] |
N. Hayashi and P. I. Naumkin,
Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math., 11 (2009), 771-781.
|
| [12] |
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, in: Mathématiques et Applications, 26, Springer, Berlin, 1997. |
| [13] |
S. Katayama,
A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ., 39 (1999), 203-213.
|
| [14] |
S. Katayama, T. Ozawa and H. Sunagawa,
A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math., 65 (2012), 1285-1302.
|
| [15] |
Y. Kawahara and H. Sunagawa,
Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations, 251 (2011), 2549-2567.
|
| [16] |
M. Keel and T. Tao,
Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.
|
| [17] |
S. Klainerman,
Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., 38 (1985), 631-641.
|
| [18] |
H. Lindblad and A. Soffer,
A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ., 1 (2005), 77-89.
|
| [19] |
H. Lindblad and A. Soffer,
A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys., 73 (2005), 249-258.
|
| [20] |
B. Marshall, W. Strauss and S. Wainger,
Lp-Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl., 59 (1980), 417-440.
|
| [21] |
S. Masaki and H. Miyazaki, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity, preprint available at arXiv: 1612.04524. |
| [22] |
S. Masaki, H. Miyazaki and K. Uriya, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity in three space dimension, preprint. |
| [23] |
S. Masaki and J. Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg-de Vries equation, to appear in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, preprint available at arXiv: 1602.05331. |
| [24] |
S. Masaki and J. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two dimensions, to appear in Trans. AMS, preprint available at arXiv: 1612.00109. |
| [25] |
A. Matsumura,
On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976/77), 169-189.
|
| [26] |
K. Moriyama,
Normal forms and global existence of solutions to a class of cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 10 (1997), 499-520.
|
| [27] |
K. Moriyama, S. Tonegawa and Y. Tsutsumi,
Wave operators for the nonlinear Schrödinger equation with a nonlinearity of low degree in one or two space dimensions, Commun. Contemp. Math., 5 (2003), 983-996.
|
| [28] |
T. Ozawa, K. Tsutaya and Y. Tsutsumi,
Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z., 222 (1996), 341-362.
|
| [29] |
H. Pecher,
Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.
|
| [30] |
H. Pecher,
Low energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal., 63 (1985), 101-122.
|
| [31] |
J. Shatah,
Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., 38 (1985), 685-696.
|
| [32] |
A. Shimomura and S. Tonegawa,
Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions, Differential Integral Equations, 17 (2004), 127-150.
|
| [33] |
W. A. Strauss,
Nonlinear scattering theory at low energy, J. Funct. Anal., 41 (1981), 110-133.
|
| [34] |
H. Sunagawa,
Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan, 58 (2006), 379-400.
|
| [35] |
H. Sunagawa,
Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations, 18 (2005), 481-494.
|
| [36] |
K. Yajima,
Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), 415-426.
|
| [1] |
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 |
| [2] |
Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973 |
| [3] |
Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085 |
| [4] |
Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889 |
| [5] |
Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753 |
| [6] |
Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013 |
| [7] |
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 |
| [8] |
Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043 |
| [9] |
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 |
| [10] |
Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233 |
| [11] |
Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 |
| [12] |
Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071 |
| [13] |
Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233 |
| [14] |
Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315 |
| [15] |
Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048 |
| [16] |
Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679 |
| [17] |
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 |
| [18] |
Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251 |
| [19] |
Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359 |
| [20] |
Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]