July 2018, 17(4): 1595-1611. doi: 10.3934/cpaa.2018076

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

* Corresponding author

Received  February 2017 Revised  July 2017 Published  April 2018

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 [23] and smooth modification of phase correction by Ginibre and Ozawa [6].

Citation: 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
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. DelortD. 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. KatayamaT. 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. MarshallW. 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. MoriyamaS. 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. OzawaK. 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. DelortD. 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. KatayamaT. 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. MarshallW. 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. MoriyamaS. 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. OzawaK. 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753

[6]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-22. doi: 10.3934/dcdsb.2017215

[7]

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

[8]

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

[9]

Yang Han. On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 233-242. doi: 10.3934/dcds.2005.12.233

[10]

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

[11]

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

[12]

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

[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]

Daniel Bouche, Youngjoon Hong, Chang-Yeol Jung. Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1159-1181. doi: 10.3934/dcds.2017048

[15]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

[16]

Michael V. Klibanov. A phaseless inverse scattering problem for the 3-D Helmholtz equation. Inverse Problems & Imaging, 2017, 11 (2) : 263-276. doi: 10.3934/ipi.2017013

[17]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55

[18]

Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042

[19]

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

[20]

Zaihui Gan. Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1541-1554. doi: 10.3934/cpaa.2009.8.1541

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (18)
  • HTML views (51)
  • Cited by (0)

Other articles
by authors

[Back to Top]