High frequency asymptotic analysis for nonlinear PDEs of hyperbolic and dispersive type

• Project/Area Number: 22740089
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Osaka University
• Principal Investigator: SUNAGAWA Hideaki 大阪大学, 大学院・理学研究科, 准教授 (80375394)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 漸近解析 / 非線形波動 / シュレディンガー方程式 / 波動方程式 / クライン・ゴルドン方程式

Research Abstract

Nonlinear partial differential equations of hyperbolic and dispersive type are studied from the view point of high frequency asymptotic analysis. Relations between the null structure and resonant interactions are revealed for systems of nonlinear Klein-Gordon equations. Examples on small data blow-up for nonlinear Schrodinger equations are constructed, and estimates for the order of the lifespan of the blowing-up solutions are provided. The method of high frequency asymptotics are also applied to a class of nonlinear wave equations not satisfying the null condition.

Research Products

• [Journal Article] Small data blow-up for a system of nonlinear Schrodinger equations (2013)
  • Author(s): T. Ozawa and H. Sunagawa
  • Journal Title: Journal of Mathematical Analysis and Application s Volume: vol.399 Pages: 147-155
  • Peer Reviewed
• [Journal Article] Small data blow-up for a system of nonlinear Schrodinger equations (2013)
  • Author(s): Tohru Ozawa, Hideaki Sunagawa
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: ３９９ Issue: 1 Pages: 147-155
  • DOI: 10.1016/j.jmaa.2012.10.003
  • Peer Reviewed
• [Journal Article] A note on the null condition for quadratic nonlinear Klein-Gordon systems in twospace dimensions (2012)
  • Author(s): S. Katayama, T. Ozawa and H. Sunagawa
  • Journal Title: Communications on Pure and Applied Mathematics Volume: vol.65 Pages: 1285-1302
  • Peer Reviewed
• [Journal Article] A remark on the algebraic normal form method applied tothe Dirac-Klein- Gordon system in twospace dimensions (2012)
  • Author(s): M. Ikeda, A. Shimomura and H.Sunagawa
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: vol.B33 Pages: 87-96
  • Peer Reviewed
• [Journal Article] The energy decay and asymptotics for a class of semilinear wave equations in twospace dimensions (2012)
  • Author(s): S. Katayama,D. Murotani and H.Sunagawa
  • Journal Title: Journal of Evolution Equations Volume: vol.12 Pages: 891-916
  • Peer Reviewed
• [Journal Article] The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions (2012)
  • Author(s): Soichiro Katayama, Daisuke Murotani, Hideaki Sunagawa
  • Journal Title: Journal of Evolution Equations Volume: 12 Issue: 4 Pages: 891-916
  • DOI: 10.1007/s00028-012-0160-4
  • Peer Reviewed
• [Journal Article] A note on the null condition for quadratic nonlinear Klein-Gord on systems in two space dimensions (2012)
  • Author(s): 片山聡一郎、小澤徹、砂川秀明
  • Journal Title: Communications on Pure and Applied Mathematics Volume: 65 Issue: 9 Pages: 1285-1302
  • DOI: 10.1002/cpa.21392
  • Peer Reviewed
• [Journal Article] A remark on the algebraic normal form method applied to the Dirac-Klein-Gordon system in two space dimensions (2012)
  • Author(s): 池田正弘、下村明洋、砂川秀明
  • Journal Title: 数理解析研究所講究録別冊 Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] Global small amplitude solutions for twodimensional nonlinear Klein-Gordon systems in the presence of mass resonance (2011)
  • Author(s): Y. Kawahara and H. Sunagawa
  • Journal Title: Journal of Differential Equations Volume: vol.251 Pages: 2549-2567
  • Peer Reviewed
• [Journal Article] Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance (2011)
  • Author(s): 川原雄一朗、砂川秀明
  • Journal Title: Journal of Differential Equations Volume: 251 Issue: 9 Pages: 2549-2567
  • DOI: 10.1016/j.jde.2011.04.001
  • Peer Reviewed
• [Presentation] 微分型非線形シュレディンガー方程式系の零構造について (2012)
  • Author(s): 砂川秀明
  • Organizer: 第10回浜松編微分方程式研究集会
  • Place of Presentation: 静岡大学
  • Year and Date: 2012-12-19
• [Presentation] Quadratic nonlinear Schrodinger systems in the presence of mass resonance (2012)
  • Author(s): 砂川秀明
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学
  • Year and Date: 2012-12-15
• [Presentation] Small data blow-up for NLS systems (2012)
  • Author(s): H.Sunagawa
  • Organizer: China-Japan Joint Meeting on PDE at Yanji
  • Place of Presentation: 延辺大学(中国)
  • Year and Date: 2012-09-10
• [Presentation] On quadratic nonlinear Klein-Gordon systems in twospace dimensions (2011)
  • Author(s): H.Sunagawa
  • Organizer: 4th MSJ-SI: Nonlinear dynamics in partial differential equations
  • Place of Presentation: 九州大学
  • Year and Date: 2011-09-18
• [Presentation] On quadratic nonlinear Klein-Gordon systems in two space dimensions (2011)
  • Author(s): 砂川秀明
  • Organizer: The 4th MSJ-SI "Nonlinear Dynamics in Partial Differential Equations"
  • Place of Presentation: 九州大学医学部百年講堂(招待講演)
  • Year and Date: 2011-09-18
• [Presentation] 「On quadratic nonlinear Klein-Gordon systems in twospace dimensions (2011)
  • Author(s): H.Sunagawa
  • Organizer: 調和解析と非線形偏微分方程式
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-07-04
• [Presentation] On quadratic nonlinear Klein-Gordon systems in two space dimensions (2011)
  • Author(s): 砂川秀明
  • Organizer: 調和解析と非線形偏微分方程式
  • Place of Presentation: 京都大学数理解析研究所(招待講演)
  • Year and Date: 2011-07-04
• [Presentation] Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance (2010)
  • Author(s): H.Sunagawa
  • Organizer: Joint workshop on PDE at Jinhua
  • Place of Presentation: 浙江師範大学(中国)
  • Year and Date: 2010-09-27
• [Presentation] Global small amplitude solutions for two dimensionlal nonlinear Klein-Gordon systems in the presence of mass resonance (2010)
  • Author(s): 砂川秀明
  • Organizer: Joint Workshop on PDE at Jinhua
  • Place of Presentation: 浙江師範大学(中国)
  • Year and Date: 2010-09-27
• [Presentation] Small data blow-up for NLS systems
  • Author(s): Hideaki Sunagawa
  • Organizer: China-Japan Joint Meeting on PDE at Yanji
  • Place of Presentation: Yanbian University
  • Invited