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Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions

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  • 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].

    Mathematics Subject Classification: Primary: 35L71; Secondary: 35B40, 81Q05.

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  • [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. WaingerLp-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. 
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