Study of nonlinear dispersive equations in critical Sobolev spaces

• Project/Area Number: 25800077
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Shizuoka University
• Principal Investigator: Akahori Takafumi 静岡大学, 工学部, 准教授 (90437187)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 非線形シュレディンガー方程式 / 散乱問題 / 基底状態の安定性 / 非線形シュレディンガー方程式のダイナミクス / 基底状態

Outline of Final Research Achievements

I studied a combined power-type nonlinear Schrodinger equation including the energy critical exponent. The aim is to reveal the behavior of the solutions starting from a neighborhood of a ground state. As a result, I proved that for any ground state with a sufficiently small frequency, the radial solutions starting from its neighbourhood exhibit one of the following scenarios: scattering to a free solution, blowup, or trapping by the ground state. Moreover, our proof clarifies the relation between potential wells with different frequencies.

Research Products

• [Int'l Joint Research] University of Victoria(カナダ)
• [Journal Article] Blowup and scattering problems for the nonlinear Schrodinger equations (2013)
  • Author(s): Takafumi Akahori, Hayato Nawa
  • Journal Title: Kyoto Journal of Mathematics Volume: vol 53, no 3 Issue: 3 Pages: 629-672
  • DOI: 10.1215/21562261-2265914
  • Peer Reviewed
• [Journal Article] Existence of a ground state and scattering for a nonlinear Schrodinger equation with critical growth (2013)
  • Author(s): Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi, Hayato Nawa
  • Journal Title: Selecta Mathematica, new series Volume: vol 19, issue 2 Issue: 2 Pages: 545-609
  • DOI: 10.1007/s00029-012-0103-5
  • Peer Reviewed
• [Presentation] Dynamics around ground states for combined power-type nonlinear Schrodinger equations with critical growth (2016)
  • Author(s): 赤堀公史
  • Organizer: Applied math seminar
  • Place of Presentation: University of Victoria
  • Invited
• [Presentation] Global Dynamics for the nonlinear Schrodinger equation with combined power-type nonlinearities (2015)
  • Author(s): 赤堀公史
  • Organizer: Applied Math seminar
  • Place of Presentation: University of Victoria (Canada)
  • Year and Date: 2015-10-06
  • Invited
• [Presentation] Global dynamics above the ground state energy for a class on nonlinear Schrodinger equations with critical growth -small frequency case- (2015)
  • Author(s): 赤堀公史
  • Organizer: 第12回 青葉山勉強会
  • Place of Presentation: 東北大学大学院情報科学研究科 (宮城県仙台市)
  • Year and Date: 2015-06-30
  • Invited
• [Presentation] Global dynamics above the ground state energy for a class on nonlinear Schrodinger equations with critical growth -small frequency case- (2015)
  • Author(s): 赤堀公史
  • Organizer: 線形および非線形分散型方程式の研究
  • Place of Presentation: 京都大学数理解析研究所 (京都府京都市)
  • Year and Date: 2015-05-19
  • Invited
• [Presentation] On nonlinear Schrodinger equations with critical growth (2014)
  • Author(s): 赤堀公史
  • Organizer: 線形および非線形分散型方程式の研究
  • Place of Presentation: 京都府京都市 京都大学数理解析研究所
  • Year and Date: 2014-05-20