Study of behavior of solutions to nonlinear dispersive equations

• Project/Area Number: 22740092
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Shizuoka University (2011-2012); Ehime University (2010)
• Principal Investigator: AKAHORI Takafumi 静岡大学, 工学部, 准教授 (90437187)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 関数方程式 / 非線形シュレディンガー方程式 / 散乱問題 / 基底状態 / 爆発問題 / 非線形分散型方程式

Research Abstract

I have studied nonlinear dispersive equations. Especially, I studied nonlinear Schrodinger equations containing the energy-critical term, and proved the followings:
(1) Existence of unstable ground state in four dimensions and higher.
(2) Nonexistence of ground state in three dimension.
(3) Classification of the behavior of solutions below the ground state energy, in view of scattering and blowup

Research Products

• [Journal Article] Existence of a ground state and scattering problem for a nonlinear Schrodinger equation with critical growth (2013)
  • Author(s): 赤堀公史, Slim Ibrahim, 菊池弘明, 名和範人
  • Journal Title: Selecta Mathematica Volume: 19巻 Pages: 545-609
  • Peer Reviewed
• [Journal Article] Scattering and blowup problems for a class of nonlinear Schrodinger equations (2012)
  • Author(s): 赤堀公史, 菊池弘明, 名和範人
  • Journal Title: Differential and Integral Equations Volume: 25巻 Pages: 1075-1118
  • Peer Reviewed
• [Journal Article] Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth (2012)
  • Author(s): 赤堀公史, Slim Ibrahim, 菊池弘明, 名和範人
  • Volume: 25巻
  • Pages: 383-402
  • Peer Reviewed
• [Journal Article] Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth (2012)
  • Author(s): 赤堀公史, Slim Ibrahim, 菊池弘明, 名範人
  • Journal Title: Differential and Integral Equations Volume: 25 Pages: 383-402
  • Peer Reviewed
• [Presentation] Remarks on the energy-critical nonlinear Schrodinger equation, UVic one day seminar on Dispersive PDEs (2013)
  • Author(s): 赤堀公史
  • Organizer: Human and Social Development Building
  • Place of Presentation: University of Victoria, Canada
  • Year and Date: 2013-03-08
• [Presentation] Remarks on the energy-critical nonlinear Schrodinger equation (2013)
  • Author(s): 赤堀公史
  • Organizer: 松山解析セミナー2013
  • Place of Presentation: 愛媛大学理学部
  • Year and Date: 2013-02-09
• [Presentation] Remarks on the energy-critical nonlinear Schrodinger equation (2013)
  • Author(s): 赤堀公史
  • Organizer: 松山解析セミナー2013
  • Place of Presentation: 愛媛大学理学部（愛媛県）
  • Invited
• [Presentation] Scattering for nonlinear Schrodinger equations at critical regularity (2012)
  • Author(s): 赤堀公史
  • Organizer: 非線形分散型方程式における最近の進展
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-05-21
• [Presentation] Scattering for nonlinear Schrodinger equations at critical regularity (2012)
  • Author(s): 赤堀公史
  • Organizer: 非線形分散型方程式における最近の進展
  • Place of Presentation: 京都大学数理解析研究所(京都府)
• [Presentation] Scattering and blowup problems for a class of nonlinear Schrodinger equations (2011)
  • Author(s): 赤堀公史
  • Organizer: 非線形双曲型および分散型方程式の研究
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-05-24
• [Presentation] Scattering problem for nonlinear Schrodinger equations (2010)
  • Author(s): 赤堀公史
  • Organizer: 広島応用解析セミナー
  • Place of Presentation: 広島大学工学部
  • Year and Date: 2010-08-23
• [Presentation] Scattering problem for the nonlinear Schrodinger equation with combined power nonlinearities (2010)
  • Author(s): 赤堀公史
  • Organizer: Harmonic Analysis and Partial Differential Equations
  • Place of Presentation: Seoul National University
  • Year and Date: 2010-08-20
• [Presentation] 非線形シュレディンガー方程式の爆発問題について (2010)
  • Author(s): 赤堀公史
  • Organizer: 非線形波動・分散型方程式の研究
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-05-25