On study of evolution equations with hyperbolic properties

• Project/Area Number: 19340030
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka University
• Principal Investigator: HAYASHI Nakao 大阪大学, 大学院・理学研究科, 教授 (30173016)
• Co-Investigator(Kenkyū-buntansha): NISHITANI Tatsuo 大阪大学, 大学院・理学研究科, 教授 (80127117) ; DOI Shinichi 大阪大学, 大学院・理学研究科, 教授 (00243006) ; KUBO Hideo 大阪大学, 大学院・理学研究科, 准教授 (50283346)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥13,000,000 (Direct Cost: ¥10,000,000、Indirect Cost: ¥3,000,000); Fiscal Year 2010: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2009: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000); Fiscal Year 2008: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2007: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
• Keywords: 関数方程式 / シュレディンガー方程式 / クライン-ゴルドン方程式 / 漸近的振る舞い / 共鳴現象 / 散乱問題 / 臨界冪非線形項 / 散乱作用素 / 修正散乱作用素 / 非線形Klein-Gordon方程式 / 非線形Schredinger方程式 / 双曲型方程式 / 分散型方程式 / シュレデインガー方程式 / クラインーゴルドン方程式 / 固有振動数 / 非線形問題 / 非線形Klein-Gordon / 非線形Schrodinger / 臨界冪 / 消散項 / 熱方程式

Research Abstract

We studied nonlinear Schrodinger equations, nonlinear Klein-Gordon equations and their systems. Time decay and asymptotic behavior of solutions were shown. We applied these results to show existence of wave or modified wave operators. In the case of critical nonlinearities, it was shown that main terms of solutions can be represented through nonlinearities clearly.

Research Products

• [Journal Article] Small data scattering for a system of nonlinear Schrodinger equations (2011)
  • Author(s): N. Hayashi, C. Li and P. I. Naumkin
  • Journal Title: Differential Equations and Applications Volume: 3, no.3 Pages: 415-426
  • Peer Reviewed
• [Journal Article] Global existence for the cubic nonlinear Schredinger equation in lower order Sobolev spaces (2011)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Differential Integral Equations Volume: 24 Pages: 801-828
  • Peer Reviewed
• [Journal Article] On a system of nonlinear Schrodinger equations in 2d (2011)
  • Author(s): N. Hayashi, C. Li and P. I. Naumkin
  • Journal Title: Differential Integral Equations Volume: 24 Pages: 417-434
  • Peer Reviewed
• [Journal Article] Wave operator for the system of the Dirac-Klein-Gordon equations (2011)
  • Author(s): N. Hayashi, M. Ikeda and P. I. Naumkin
  • Journal Title: Math. Methods Appl. Sciences Volume: 34, no.8 Pages: 896-910
  • Peer Reviewed
• [Journal Article] Asymptotics for nonlinear heat equations (2011)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Nonlinear analysis Volume: 74, no.5 Pages: 1585-1595
  • Peer Reviewed
• [Journal Article] Subcritical quadratic nonlinear Schrodinger equation (2011)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Communications in Contemporary Mathematics Volume: 13 Pages: 969-1007
  • Peer Reviewed
• [Journal Article] Quadratic derivative nonlinear Schrodinger equations in two space dimensions (2011)
  • Author(s): Nakao Hayashi
  • Journal Title: Nonlinear Differential Equations and Applications Volume: 18 Pages: 329-355
  • Peer Reviewed
• [Journal Article] Small data scattering for a system of nonlinear Schrodinger equations (2011)
  • Author(s): Nakao Hayashi
  • Journal Title: Differential Equations and Applications Volume: 3 Pages: 415-426
  • Peer Reviewed
• [Journal Article] Subcritical quadratic nonlinear Schrodinger equation (2011)
  • Author(s): Nakao Hayashi
  • Journal Title: Commun.Contemporary Math. Volume: 13 Pages: 969-1007
  • Peer Reviewed
• [Journal Article] Global existence for the cubic nonlinear Schrodinger equation in lower order Sobolev spaces (2011)
  • Author(s): Nakao Hayashi
  • Journal Title: Differential Integral Equations Volume: 24 Pages: 801-828
  • Peer Reviewed
• [Journal Article] Analyticity for higher order nonlinear dispersive equations, GAKUTO, International Series, Mathematical Sciences and Applications (2010)
  • Author(s): N. Hayashi, K. Tomoeda and T. Yanagisawa
  • Journal Title: Current Advances in Nonlinear Analysis and Related Topics Volume: 32 Pages: 111-130
  • Peer Reviewed
• [Journal Article] The initial value problem for the quadratic nonlinear Klein-Gordon equation (2010)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Advances in Mathematical Physics Pages: 504324-504324
  • Peer Reviewed
• [Journal Article] Benjamin-Ono equation on a half-line (2010)
  • Author(s): N. Hayashi and E. Kaikina
  • Journal Title: International Journal of Mathematics and Mathematical Sciences Pages: 714534-714534
  • Peer Reviewed
• [Journal Article] The initial value problem for the quadratic nonlinear Klein-Gordon equation (2010)
  • Author(s): Nakao Hayashi
  • Journal Title: Advances in Mathematical Physics Volume: Article ID 504324
  • Peer Reviewed
• [Journal Article] Benjamin-Ono equation on a half-line (2010)
  • Author(s): Nakao Hayashi
  • Journal Title: International Journal of Mathematics and Mathematical Sciences Volume: Article ID 714534
  • Peer Reviewed
• [Journal Article] Analyticity for higher order nonlinear dispersive equations (2010)
  • Author(s): Nakao Hayashi
  • Journal Title: GAKUTO, International Series, Mathematical Sciences and Applications Volume: 32 Pages: 111-130
  • Peer Reviewed
• [Journal Article] Wave operator for the system of the Dirac-Klein-Gordon equations (2010)
  • Author(s): Nakao Hayashi
  • Journal Title: Math.Methods Appl.Sciences Volume: 34 Pages: 896-910
  • Peer Reviewed
• [Journal Article] Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions (2009)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Nonlinear Analysis Volume: 71, no.9 Pages: 3826-3833
  • Peer Reviewed
• [Journal Article] Asymptotics of odd solutions of quadratic nonlinear Schrodinger equations (2009)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: J. Math. Anal. Appl. Volume: 359, no.1 Pages: 146-158
  • Peer Reviewed
• [Journal Article] Asymptotics of odd solutions for cubic nonlinear Schrodinger equations (2009)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: J. Differential Equations Volume: 246, no.4 Pages: 1703-1722
  • Peer Reviewed
• [Journal Article] Scattering Operator for Nonlinear Klein-Gordon Equations (2009)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Communications in Contemporary Mathematics Volume: 11, no.5 Pages: 771-178
  • Peer Reviewed
• [Journal Article] Final state problem for the cubic nonlinear Klein-Gordon equation (2009)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: J. Math. Phys. Volume: 50, no.10 Pages: 103511-103511
  • Peer Reviewed
• [Journal Article] Final state problem for the cubic nonlinear Klein-Gordon equation (2009)
  • Author(s): N. Hayashi, P. I. Naumkin
  • Journal Title: J. Math. Phys. vol. 50, no. 10 Pages: 103511-14
  • Peer Reviewed
• [Journal Article] Scattering Operator for Nonlinear Klein-Gordon Equations (2009)
  • Author(s): N.Hayashi
  • Journal Title: Commum.Contemp.Math. 11 Pages: 771-781
  • Peer Reviewed
• [Journal Article] Asymptotics of odd solutions for cubic nonlinear Schredinger equations (2009)
  • Author(s): N.Hayashi
  • Journal Title: J.Differential Equations 246 Pages: 1703-1722
  • Peer Reviewed
• [Journal Article] Asymptotics of odd solutions of quadratic nonlinear Schredinger equations (2009)
  • Author(s): N.Hayashi
  • Journal Title: J.Math.Anal.Appl. 359 Pages: 146-158
  • Peer Reviewed
• [Journal Article] Nonlinear scattering for a system of one dimensional nonlinear Klein-Gordon equations (2008)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Hokkaido Math. J. Volume: 37 Pages: 647-667
  • Peer Reviewed
• [Journal Article] Nonlinear scattering for a system of nonlinear Klein-Gordon equations (2008)
  • Author(s): N. Hayashi, P. I. Naumkin and Ratno Bagus Edy Wibowo
  • Journal Title: J. Math. Phys. Volume: 49, no.10 Pages: 103501-103501
  • Peer Reviewed
• [Journal Article] On the Schrodinger equation with dissipative nonlinearities of derivative type (2008)
  • Author(s): N. Hayashi, P. I. Naumkin and H. Sunagawa
  • Journal Title: SIAM J. Math. Anal. Volume: 40 Pages: 278-291
  • Peer Reviewed
• [Journal Article] Modified wave operators for nonlinear Schrodinger equations with subcritical dissipative nonlinearities (2008)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Inverse Problems and Imaging Volume: 1, no.2 Pages: 391-398
  • Peer Reviewed
• [Journal Article] The initial value problem and asymptotics of solutions for the cubic nonlinear Klein-Gordon equation (2008)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: Zeitschrift fur Angewandte Math. und Physik Volume: 59, no.6 Pages: 1002-1028
  • Peer Reviewed
• [Journal Article] Scattering operator for the nonlinear Klein-Gordon equations in higher space dimensions (2008)
  • Author(s): N. Hayashi and P. I. Naumkin
  • Journal Title: J. Differential Equations Volume: 244 Pages: 188-199
  • Peer Reviewed
• [Journal Article] Asymptotic properties of solutions to dispersive equation of Schredinger type (2008)
  • Author(s): N. Hayashi, P. I. Naumkin
  • Journal Title: J. Math. Soc. Japan vol. 60 Pages: 631-652
  • Peer Reviewed
• [Journal Article] The initial value problem and asymptotics of solutions for the cubic nonlinear Klein-Gordon equation (2008)
  • Author(s): N. Hayashi, P. I. Naumkin
  • Journal Title: Zeitschrift fur Angewandte Mathematik und Physik vol. 59, no. 6 Pages: 1002-1028
  • Peer Reviewed
• [Journal Article] Asymptotic properties of solutions to dispersive equation of Schrodinger type (2008)
  • Author(s): N. Hayashi
  • Journal Title: J. Math. Soc. Japan 60 Pages: 631-652
  • NAID: 10024331840
  • Peer Reviewed
• [Journal Article] On the Schrodinger equation with dissipative nonlinearities of derivative type (2008)
  • Author(s): N. Hayashi
  • Journal Title: SIAM J. Math. Anal. 40 Pages: 278-291
  • Peer Reviewed
• [Journal Article] Nonlinear scattering for a system of nonlinear Klein-Gordon equations (2008)
  • Author(s): N. Hayashi
  • Journal Title: J. Math. Phys. 49 Pages: 103501-24
  • Peer Reviewed
• [Journal Article] Scattering operator for the nonlinear Klein-Gordon equations in higher space dimensions (2008)
  • Author(s): N. Hayashi
  • Journal Title: Journal Differential Equations 244 Pages: 188-199
  • Peer Reviewed
• [Journal Article] Modified wave operators for nonlinear Schrodinger equations with subcritical dissipative nonlinearities (2007)
  • Author(s): N. Hayashi
  • Journal Title: Inverse Problems and Imaging 1 Pages: 391-398
  • Peer Reviewed
• [Journal Article] Subcritical nonlinear heat equation (2007)
  • Author(s): N. Hayashi
  • Journal Title: Journal Differential Equations 238 Pages: 366-380
  • Peer Reviewed
• [Presentation] Quadratic nonlinear Klein-Gordon equation in 2d (2012)
  • Author(s): 林仲夫
  • Organizer: 日本数学会年会
  • Place of Presentation: 東京理科大学
  • Year and Date: 2012-03-29
• [Presentation] Quadratic nonlinear Klein-Gordon equations in 1D (2011)
  • Author(s): 林仲夫
  • Organizer: The 8th East Asia PDE Conference
  • Place of Presentation: Posco International Center, Postech, Pohang(韓国)
  • Year and Date: 2011-12-21
• [Presentation] Quadratic nonlinear Klein-Gordon equations in 1D (2011)
  • Author(s): 林仲夫
  • Organizer: The 8th East Asia PDE Conference
  • Place of Presentation: Posco International Center, Postech, Pohang(韓国)(招待講演)
  • Year and Date: 2011-12-21
• [Presentation] Bilinear estimates and Nonlinear Klein-Gordon equations (2011)
  • Author(s): 林仲夫
  • Organizer: The 4th MSJ-SI Mathematical Society of Japan, Seasonal Institute
  • Place of Presentation: 九州大学
  • Year and Date: 2011-09-19
• [Presentation] Bilinear estimates and Nonlinear Klein-Gordon equations (2011)
  • Author(s): 林仲夫
  • Organizer: The 4th MSJ-SI Mathematical Society of Japan, Seasonal Institute
  • Place of Presentation: 九州大学(招待講演)
  • Year and Date: 2011-09-19
• [Presentation] 双線形作用素の評価と非線形Klein-Gordon方程式への応用 (2011)
  • Author(s): 林仲夫
  • Organizer: 大阪市大・大阪府大合同「第2回南大阪応用数学セミナー」
  • Place of Presentation: 大阪市立大
  • Year and Date: 2011-07-02
• [Presentation] 双線形作用素の評価と非線形Klein-Gordon方程式への応用 (2011)
  • Author(s): 林仲夫
  • Organizer: 大阪市大・大阪府大合同「第2回南大阪応用数学セミナー」
  • Place of Presentation: 大阪市立大学(招待講演)
  • Year and Date: 2011-07-02
• [Presentation] A bilinear estimate and its application to a quadratic nonlinear Klein-Gordon equation in two space dimensions (2011)
  • Author(s): 林仲夫
  • Organizer: 阪大微分方程式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2011-05-27
• [Presentation] Nonexistence of scattering states for nonlinear dispersive equations (2010)
  • Author(s): 林仲夫
  • Organizer: 偏微分方程式と数理物理学
  • Place of Presentation: 京都芝蘭会館
  • Year and Date: 2010-11-17
• [Presentation] Nonexistence of scattering states for nonlinear dispersive equations (2010)
  • Author(s): 林仲夫
  • Organizer: 偏微分方程式と数理物理学
  • Place of Presentation: 京都芝蘭会館(招待講演)
  • Year and Date: 2010-11-16
• [Presentation] Modified wave operator for a system of nonlinear Schrodinger equations (2010)
  • Author(s): 林仲夫
  • Organizer: 「早稲田大学非線形偏微分方程式研究所」設立研究集会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-11-10
• [Presentation] Modified wave operator for a system of nonlinear Schrodinger equations (2010)
  • Author(s): 林仲夫
  • Organizer: 「早稲田大学非線形偏微分方程式研究所」設立研究集会ワークショップ
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2010-11-10
• [Presentation] 非線形分散型方程式の散乱問題I,II (2010)
  • Author(s): 林仲夫
  • Organizer: 「早稲田大学非線形偏微分方程式研究所」設立研究集会ミニコース
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2010-11-09
• [Presentation] Modified wave operator for a system of NLS equations in 2D (2010)
  • Author(s): 林仲夫
  • Organizer: Joint workshop on PDE at Jinhua
  • Place of Presentation: Zhejiang Normal University(中国)
  • Year and Date: 2010-09-27
• [Presentation] Modified wave operator for a system of NLS eguations in 2D (2010)
  • Author(s): 林仲夫
  • Organizer: Joint workshop on PDE at Jinhua
  • Place of Presentation: Zhejiang Normal University(中国)(招待講演)
  • Year and Date: 2010-09-27
• [Presentation] Schrodinger方程式系の解の時間減衰評価 (2010)
  • Author(s): 李春花, 林仲夫
  • Organizer: 2010日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] Schrodinger方程式系の解の時間減衰評価 (2010)
  • Author(s): 李春花, 林仲夫
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] Time decay of solutions to a system of nonlinear Schrodinger equations (2010)
  • Author(s): 林仲夫
  • Organizer: 第六回非線型の諸問
  • Place of Presentation: 山口大学
  • Year and Date: 2010-09-17
• [Presentation] Time decay of solutions to a system of nonlinear Schrodinger equations (2010)
  • Author(s): 林仲夫
  • Organizer: 第六回非線型の諸問題
  • Place of Presentation: 山口大学(招待講演)
  • Year and Date: 2010-09-16
• [Presentation] Asymptotic behavior of solutions to nonlinear Schrodinger equations (2010)
  • Author(s): 林仲夫
  • Organizer: The 35th Sapporo Symposium on Partial Differential Equations
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-08-24
• [Presentation] Asymptotic behavior of solutions to nonlinear Schrodinger equations (2010)
  • Author(s): 林仲夫
  • Organizer: The 35th Sapporo Symposium on Partial Differential Equations
  • Place of Presentation: 北海道大学(招待講演)
  • Year and Date: 2010-08-24
• [Presentation] On a system of nonlinear Schrodinger equations in 2d (2010)
  • Author(s): 林仲夫
  • Organizer: 応用解析研究会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-05-15
• [Presentation] On a system of nonlinear Schrodinger equations in 2d (2010)
  • Author(s): 林仲夫
  • Organizer: 応用解析研究会
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2010-05-15
• [Presentation] 非線形Schrodinger型方程式 (2009)
  • Author(s): 林仲夫
  • Organizer: 奈良女子大学数学談話会
  • Place of Presentation: 奈良女子大学
  • Year and Date: 2009-11-30
• [Presentation] 空間2次元における2次の非線形クライン・ゴルドン方程式に対する波動作用素について (2009)
  • Author(s): 林仲夫, P. Naumkin, 利根川聡
  • Organizer: 日本数学会秋季大会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] Dirac-Klein-Gordon方程式系の波動作用素 (2009)
  • Author(s): 池田正弘, 林仲夫
  • Organizer: 日本数学会秋季大会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] Direac-Klein-Gordon方程式系の波動作用素 (2009)
  • Author(s): 池田正弘, 林仲夫
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] 空間2次元における2次の非線形クライン・ゴルドン方程式に対する (2009)
  • Author(s): 林仲夫, P.Naumkin, 利根川聡
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] 非線形Klein-Gordon方程式の散乱問題 (2009)
  • Author(s): 林仲夫
  • Organizer: 日本数学会,解析学賞受賞特別講演
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-28
• [Presentation] 非線形 Klein-Gordon 方程式の散乱問題 (2009)
  • Author(s): 林仲夫
  • Organizer: 日本数学会, 解析学賞受賞特別講演
  • Place of Presentation: 東京大学
  • Year and Date: 2009-03-26
• [Presentation] Dirac-Klein-Gordon方程式の散乱問題について (2009)
  • Author(s): 林仲夫
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2009-01-30
• [Presentation] 非線形分散型方程式の漸近解析 (2009)
  • Author(s): 林仲夫
  • Organizer: 九州非線形偏微分方程式・九州冬の学校
  • Place of Presentation: 福岡大学
• [Presentation] Dirac-Klein-Gordon方程式系の散乱問題 (2008)
  • Author(s): B. E. W. Ratno, 林仲夫
  • Organizer: 日本数学会秋季大会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-24
• [Presentation] Dirac-Klein-Gordon方程式系の散乱問題 (2008)
  • Author(s): B. E. W. Ratno 林 仲夫
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-24
• [Presentation] Dirac-Klein-Gordon方程式系の散乱問題 (2008)
  • Author(s): 林 仲夫
  • Organizer: 阪大微分方程式セミナー
  • Place of Presentation: 大阪大学
  • Year and Date: 2008-05-30
• [Presentation] 非線形Klein-Gordon方程式系の散乱問題 (2008)
  • Author(s): 林仲夫
  • Organizer: Paris-Sud PDEセミナー
  • Place of Presentation: パリ南大学(フランス)
  • Year and Date: 2008-03-11
• [Presentation] Scattering problem for nonlinear Klein-Gordon equations (2007)
  • Author(s): Hayashi Nakao
  • Organizer: International Conference on Applied Partial Differential Equations
  • Place of Presentation: East Chaina Normal University, Shanghai, Chaina
  • Year and Date: 2007-07-05