On small data scattering with cubic convolution nonlinearity

Kiyoshi MOCHIZUKI

doi:10.2969/jmsj/04110143

Kiyoshi MOCHIZUKI

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JOURNAL FREE ACCESS

1989 Volume 41 Issue 1 Pages 143-160

DOI https://doi.org/10.2969/jmsj/04110143

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Published: 1989 Received: September 07, 1987 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Takeyuki Hida on his sixtieth birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976.
2) Ph. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344.
3) J. P. Dias and M. Figueira, Conservation laws and time decay of some nonlinear Schrödinger-Hartree equations and systems, J. Math. Anal. Appl., 84 (1981), 486-508.
4) J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.
5) R. T. Glassey, Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations, Comm. Math. Phys., 53 (1977), 9-18.
6) E. Gross, Quantum theory of interacting bosons, Ann. Physics, 9 (1960), 292-324.
7) N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincaré, Phys. Théor., 46 (1987), 187-213.
8) N. Hayashi and T. Ozawa, Scattering theory in the weighted L²(Rⁿ) spaces for some Schrödinger equations, preprint, 1987.
9) L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 1983.
10) P. L. Lions, The Choquard equation and related equations, J. Nonlinear Anal. Theory, Methods, Appl., 4 (1980), 1073-1079.
11) S. Matsumoto and K. Mochizuki, On blowing up of solutions of Schrödinger equations with cubic convolution, preprint, 1987.
12) G. P. Menzala, On regular solutions of a nonlinear equations of Choquard's type, Proc. Roy. Soc. Edinburgh, Sect. A, 86 (1980), 291-301.
13) G. P. Menzala and W. A. Strauss, On a wave equation with cubic convolution, J. Differential Equations, 43 (1982), 93-105.
14) K. Mochizuki and T. Motai, On small data scattering for some nonlinear wave equations, Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations-, 1986, pp. 543-560.
15) H. Pecher, L^p-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen I, Math. Z., 150 (1976), 159-183.
16) H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.
17) W. A. Strauss, Nonlinear scattering theory at law energy, J. Funct. Anal., 41 (1981), 110-133, Sequel, 43 (1981), 281-293.

Right : [1] J. Bergh and J. Löfström, Interpolation Spaces, Springer, 1976.
[2] Ph. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344.
[3] J. P. Dias and M. Figueira, Conservation laws and time decay of some nonlinear Schrödinger-Hartree equations and systems, J. Math. Anal. Appl., 84 (1981), 486-508.
[4] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.
[5] R. T. Glassey, Asymptotic behavior of solutions to certain nonlinear Schrödinger-Hartree equations, Comm. Math. Phys., 53 (1977), 9-18.
[6] E. Gross, Quantum theory of interacting bosons, Ann. Physics, 9 (1960), 292-324.
[7] N. Hayashi and Y. Tsutsumi, Scattering theory for Hartree type equations, Ann. Inst. H. Poincaré, Phys. Théor., 46 (1987), 187-213.
[8] N. Hayashi and T. Ozawa, Scattering theory in the weighted L²(Rⁿ) spaces for some Schrödinger equations, preprint, 1987.
[9] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer, 1983.
[10] P. L. Lions, The Choquard equation and related equations, J. Nonlinear Anal. Theory, Methods, Appl., 4 (1980), 1073-1079.
[11] S. Matsumoto and K. Mochizuki, On blowing up of solutions of Schrödinger equations with cubic convolution, preprint, 1987.
[12] G. P. Menzala, On regular solutions of a nonlinear equations of Choquard's type, Proc. Roy. Soc. Edinburgh, Sect. A, 86 (1980), 291-301.
[13] G. P. Menzala and W. A. Strauss, On a wave equation with cubic convolution, J. Differential Equations, 43 (1982), 93-105.
[14] K. Mochizuki and T. Motai, On small data scattering for some nonlinear wave equations, Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations-, 1986, pp. 543-560.
[15] H. Pecher, L^p-Abschätzungen und klassische Lösungen für nichtlineare Wellengleichungen I, Math. Z., 150 (1976), 159-183.
[16] H. Pecher, Nonlinear small data scattering for the wave and Klein-Gordon equation, Math. Z., 185 (1984), 261-270.
[17] W. A. Strauss, Nonlinear scattering theory at law energy, J. Funct. Anal., 41 (1981), 110-133, Sequel, 43 (1981), 281-293.

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