Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions
HTML articles powered by AMS MathViewer
- by Satoshi Masaki and Jun-ichi Segata PDF
- Trans. Amer. Math. Soc. 370 (2018), 8155-8170 Request permission
Abstract:
In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: $(\Box +1)u=\lambda |u|u$, $t\in \mathbb {R}$, $x\in \mathbb {R}^{2}$, where $\Box =\partial _{t}^{2}-\Delta$ is d’Alembertian. For a given asymptotic profile $u_{\mathrm {ap}}$, we construct a solution $u$ to (NLKG) which converges to $u_{\mathrm {ap}}$ as $t\to \infty$. 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 Fourier series expansion of the nonlinearity.References
- Philip Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z. 186 (1984), no. 3, 383–391. MR 744828, DOI 10.1007/BF01174891
- Philip Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations 56 (1985), no. 3, 310–344. MR 780495, DOI 10.1016/0022-0396(85)90083-X
- Jean-Marc Delort, Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 1, 1–61 (French, with English and French summaries). MR 1833089, DOI 10.1016/S0012-9593(00)01059-4
- Jean-Marc Delort, Daoyuan Fang, and Ruying Xue, Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal. 211 (2004), no. 2, 288–323. MR 2056833, DOI 10.1016/j.jfa.2004.01.008
- Vladimir Georgiev and Sandra Lucente, Weighted Sobolev spaces applied to nonlinear Klein-Gordon equation, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 21–26 (English, with English and French summaries). MR 1703291, DOI 10.1016/S0764-4442(99)80454-6
- V. Georgiev and B. Yardanov, Asymptotic behavior of the one dimensional Klein–Gordon equation with a cubic nonlinearity, preprint (1996).
- J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 43 (1985), no. 4, 399–442 (English, with French summary). MR 824083
- Robert T. Glassey, On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc. 182 (1973), 187–200. MR 330782, DOI 10.1090/S0002-9947-1973-0330782-7
- Nakao Hayashi and Pavel I. Naumkin, The initial value problem for the cubic nonlinear Klein-Gordon equation, Z. Angew. Math. Phys. 59 (2008), no. 6, 1002–1028. MR 2457221, DOI 10.1007/s00033-007-7008-8
- Nakao Hayashi and Pavel I. Naumkin, Final state problem for the cubic nonlinear Klein-Gordon equation, J. Math. Phys. 50 (2009), no. 10, 103511, 14. MR 2572684, DOI 10.1063/1.3215980
- Nakao Hayashi and Pavel I. Naumkin, Scattering operator for nonlinear Klein-Gordon equations, Commun. Contemp. Math. 11 (2009), no. 5, 771–781. MR 2561936, DOI 10.1142/S0219199709003582
- Nakao Hayashi and Pavel I. Naumkin, Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions, Nonlinear Anal. 71 (2009), no. 9, 3826–3833. MR 2536291, DOI 10.1016/j.na.2009.02.041
- Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
- Soichiro Katayama, A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension, J. Math. Kyoto Univ. 39 (1999), no. 2, 203–213. MR 1709289, DOI 10.1215/kjm/1250517908
- Soichiro Katayama, Tohru Ozawa, and Hideaki Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, Comm. Pure Appl. Math. 65 (2012), no. 9, 1285–1302. MR 2954616, DOI 10.1002/cpa.21392
- Yuichiro Kawahara and Hideaki Sunagawa, Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance, J. Differential Equations 251 (2011), no. 9, 2549–2567. MR 2825340, DOI 10.1016/j.jde.2011.04.001
- Sergiu Klainerman, Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math. 38 (1985), no. 5, 631–641. MR 803252, DOI 10.1002/cpa.3160380512
- Hans Lindblad and Avy Soffer, A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ. 2 (2005), no. 1, 77–89. MR 2134954, DOI 10.1142/S0219891605000385
- Hans Lindblad and Avy Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys. 73 (2005), no. 3, 249–258. MR 2188297, DOI 10.1007/s11005-005-0021-y
- Bernard Marshall, Walter Strauss, and Stephen Wainger, $L^{p}-L^{q}$ estimates for the Klein-Gordon equation, J. Math. Pures Appl. (9) 59 (1980), no. 4, 417–440. MR 607048
- S. Masaki and H. Miyazaki, Long range scattering for nonlinear Schrödinger equations with critical homogeneous nonlinearity, preprint, arXiv:1612.04524.
- Satoshi Masaki and Jun-ichi Segata, Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 2, 283–326. MR 3765544, DOI 10.1016/j.anihpc.2017.04.003
- Akitaka Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 1, 169–189. MR 0420031, DOI 10.2977/prims/1195190962
- Kazunori Moriyama, Satoshi Tonegawa, and Yoshio 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), no. 6, 983–996. MR 2030566, DOI 10.1142/S021919970300121X
- Kazunori 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), no. 3, 499–520. MR 1744859
- Kenji Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions $1$ and $2$, J. Funct. Anal. 169 (1999), no. 1, 201–225. MR 1726753, DOI 10.1006/jfan.1999.3503
- Tohru Ozawa, Kimitoshi Tsutaya, and Yoshio Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions, Math. Z. 222 (1996), no. 3, 341–362. MR 1400196, DOI 10.1007/PL00004540
- Hartmut Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984), no. 2, 261–270. MR 731347, DOI 10.1007/BF01181697
- Hartmut Pecher, Low energy scattering for nonlinear Klein-Gordon equations, J. Funct. Anal. 63 (1985), no. 1, 101–122. MR 795519, DOI 10.1016/0022-1236(85)90100-4
- Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256, DOI 10.1002/cpa.3160380516
- Akihiro Shimomura and Satoshi Tonegawa, Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions, Differential Integral Equations 17 (2004), no. 1-2, 127–150. MR 2035499
- Walter A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis 41 (1981), no. 1, 110–133. MR 614228, DOI 10.1016/0022-1236(81)90063-X
- Hideaki Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan 58 (2006), no. 2, 379–400. MR 2228565
- Hideaki Sunagawa, Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential Integral Equations 18 (2005), no. 5, 481–494. MR 2136975
- Kenji Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), no. 3, 415–426. MR 891945
Additional Information
- Satoshi Masaki
- Affiliation: Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, Toyonaka Osaka, 560-8531, Japan
- MR Author ID: 823235
- Email: masaki@sigmath.es.osaka-u.ac.jp
- Jun-ichi Segata
- Affiliation: Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
- Email: segata@m.tohoku.ac.jp
- Received by editor(s): November 25, 2016
- Received by editor(s) in revised form: April 17, 2017
- Published electronically: July 5, 2018
- Additional Notes: The first author was partially supported by the Sumitomo Foundation, Basic Science Research Projects No. 161145.
The second author was partially supported by JSPS, Grant-in-Aid for Young Scientists (A) 25707004. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 8155-8170
- MSC (2010): Primary 35L71; Secondary 35B40, 81Q05
- DOI: https://doi.org/10.1090/tran/7262
- MathSciNet review: 3852461