Scattering operator for nonlinear klein-Gordon equations

Communications in Contemporary Mathematics

Volumn 11, Issue 5, 2009, Pages 771-781

  Hayashi, Nakao ^a   Naumkin, Pavel I ^b  

^a OSAKA UNIVERSITY   (Japan)

^b INSTITUTO DE MATEMÁTICAS   (Mexico)

Author keywords

Asymptotics of solutions; Nonlinear Klein Gordon equation; Scattering operator

Indexed keywords

EID: 70350536288     PISSN: 02191997     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219199709003582     Document Type: Article

References (31)

1. P. Brenner, On space time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z. 186 (1984) 383-391.
2. P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations 56 (1985) 310-344.
3. T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrodinger equation, Comm. Math. Phys. 147 (1992) 75-100.
4. J.-M. Delort, Existence globale et comportement asymptotique pour l'́equation de Klein-Gordon quasi-lińeaire ̀a donńees petites en dimension 1, Ann. Sci. ́ Ecole Norm. Sup., (4) 34 (2001) 1-61.
5. 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.
6. V. Georgiev, Decay estimates for the Klein-Gordon equations, Comm. Partial Differential Equations 17 (1992) 1111-1139.
7. V. Georgiev and S. Lecente, Weighted Sobolev spaces applied to nonlinear Klein- Gordon equation, C. R. Acad. Sci. Paris Śer. I Math. 329 (1999) 21-26.
8. V. Georgiev and B. Yordanov, Asymptotic behavior of the one dimensional Klein- Gordon equation with a cubic nonlinearity, preprint (1996)
9. J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein- Gordon equation, Ann. Inst. H. Poincaŕe Phys. Th́eor. 41 (1985) 399-442.
10. R. T. Glassey, On the asymptotic behavior of nonlinear wave equations, Trans. Amer. Math. Soc. 182 (1973) 187-200.
11. 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.
12. S. Klainerman, Global existence of small amplitude solutions to nonlinear Klein- Gordon equations in four space-time dimensions, Commun. Pure Appl. Math. 38 (1985) 631-641.
13. H. Lindblad and A. Soffer, A remark on asymptotic completeness for the critical nonlinear Klein-Gordon equation, Lett. Math. Phys. 73(3) (2005) 249-258.
14. H. Lindblad and A. So.er, A remark on long range scattering for the nonlinear Klein-Gordon equation, J. Hyperbolic Differ. Equ. 2(1) (2005) 77-89.
15. B. Marshall, W. Strauss and S. Wainger, Lp - Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl. 59 (1980) 417-440.
16. A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci. 12 (1976/1977) 169-189.
17. 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.
18. K. Moriyama, S. Tonegawa and Y. Tsutsumi, Almost global existence of solutions for the quadratic semilinear Klein-Gordon equation in one space dimension, Funkcial. Ekvac. 40 (1997) 313-333.
19. K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169 (1999) 201-225.
20. T. Ozawa, K. Tustaya and Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlineaarity in two space dimensions, Math. Z. 222 (1996) 341-362.
21. H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z. 185 (1984) 261-270.
22. H. Pecher, Low energy scattering for Klein-Gordon equations, J. Funct. Anal. 63 (1985) 101-122.
23. J. Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Commun. Pure Appl. Math. 38 (1985) 685-696.
24. W. A. Strauss, Dispersion of low-energy waves for two conservative equations, Arch. Ration. Mech. Anal. 55 (1974) 86-92.
25. W. A. Strauss, Nonlinear scattering theory at low energy, J. Funct. Anal. 41 (1981) 110-133.
26. H. Sunagawa, On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass in one space dimension, J. Di.erential Equations 192 (2003) 308-325.
27. H. Sunagawa, Remarks on the asymptotic behavior of the cubic nonlinear Klein- Gordon equations in one space dimension, Di.erential Integral Equations 18 (2005) 481-494.
28. H. Sunagawa, Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan 58 (2006) 379-400.
29. B. X. Wang, On existence and scattering for critical and subcritical nonlinear Klein- Gordon equations in Hs, Nonlinear Anal. 31 (1998) 573-587.
30. B. X. Wang, Scattering of solutions for critical and subcritical nonlinear Klein- Gordon equations in Hs, Discrete Contin. Dyn. Syst. 5 (1999) 753-763.
31. R. Weder, Inverse scattering on the line for the nonlinear Klein-Gordon equation with a potential, J. Math. Anal. Appl. 252 (1999) 102-123.