# American Institute of Mathematical Sciences

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. 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.

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##### 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.
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