On analysis of interacting nonlocal waves

• Project/Area Number: 21540206
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Hirosaki University (2010-2013); Hokkaido University (2009)
• Principal Investigator: TSUTAYA Kimitoshi 弘前大学, 理工学研究科, 教授 (60250411)
• Co-Investigator(Kenkyū-buntansha): TAKAMURA Hiroyuki 公立はこだて未来大学, システム情報科学部, 教授 (40241781)
• Co-Investigator(Renkei-kenkyūsha): NAKAMURA Makoto 山形大学, 理学部, 教授 (70312634) ; MACHIHARA Shuji 埼玉大学, 理学部, 准教授 (20346373)
• Project Period (FY): 2009-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2013: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2012: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 波動方程式 / ディラック方程式 / ハートリータイプ / 散乱問題 / 時間大域解 / 漸近挙動 / ハートリー型 / 重み付き関数 / 時間大域解存在 / 散乱理論

Research Abstract

We studied the Cauchy and scattering problems for the wave equation of Hartree type. We obtained results on the existence of global solutions, blow-up in finite time, the existence of scattering operators and no scattering. We showed the existence of scattering operators under almost optimal conditions on potentials and initial data in terms of decay. We also proved the existence of scattering operators for the Dirac equation of Hartree type and the semirelativistic Hartree equation.

Research Products

• [Journal Article] Weighted estimates for a convolution appearing in the wave equation of Hartree type (2014)
  • Author(s): K. Tsutaya
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 411 Issue: 2 Pages: 719-731
  • DOI: 10.1016/j.jmaa.2013.10.021
  • Peer Reviewed
• [Journal Article] Scattering theory for the wave equation of a Hartree type in three space dimensions (2014)
  • Author(s): K. Tsutaya
  • Journal Title: Discrete and Continuous Dynamical Systems - A Volume: 34, no.5 Issue: 5 Pages: 2261-2281
  • DOI: 10.3934/dcds.2014.34.2261
  • Peer Reviewed
• [Journal Article] Scattering theory for the Dirac equation of Hartree type and the semirelativistic Hartree equation (2012)
  • Author(s): M. Nakamura and K. Tsutaya
  • Journal Title: Nonlinear Anal Volume: 75, no. 8 Issue: 8 Pages: 3531-3542
  • DOI: 10.1016/j.na.2012.01.012
  • Peer Reviewed
• [Journal Article] Scattering theory for the Dirac equation of Hartree type in 2+1 dimensions (2009)
  • Author(s): S. Machihara and K. Tsutaya
  • Journal Title: Nonlinear Analysis, Theory, Methods and Applications Volume: 71, no.12 Issue: 12 Pages: e2437-e2441
  • DOI: 10.1016/j.na.2009.05.059
  • NAID: 120002148676
  • Peer Reviewed
• [Journal Article] Scattering Theory for the Dirac Equation with a Nonlocal Term (2009)
  • Author(s): S. Machihara and K. Tsutaya
  • Journal Title: Proceedings of the Royal Society of Edinburgh Volume: 139A Pages: 867-878
  • NAID: 120006459584
  • Peer Reviewed
• [Journal Article] Scattering Theory for the Dirac Equation with a Nonlocal Term (2009)
  • Author(s): S.Machihara, K.Tsutaya
  • Journal Title: Proceedings of the Royal Society of Edinburgh 139A Pages: 867-878
  • NAID: 120006459584
  • Peer Reviewed
• [Journal Article] Scattering theory for the Dirac equation of Hartree type in 2+1 dimensions (2009)
  • Author(s): S.Machihara, K.Tsutaya
  • Journal Title: Nonlinear Analysis, Theory, Methods and Applications 71
  • NAID: 120002148676
  • Peer Reviewed
• [Presentation] On the asymptotic behavior of solutions of the wave equation of Hartree type (2014)
  • Author(s): P. Karageorgis and 津田谷公利
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学
  • Year and Date: 2014-03-16
• [Presentation] Scattering theory for a Hartree-type wave equation (2012)
  • Author(s): 津田谷公利
  • Organizer: 函館偏微分方程式研究会
  • Place of Presentation: 公立はこだて未来大学(招待講演)
  • Year and Date: 2012-01-27
• [Presentation] Global existence and asymptotic behavior of solutions to a Hartree-type wave equation (2012)
  • Author(s): K. Tsutaya
  • Organizer: 第29回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学
  • Year and Date: 2012-01-25
• [Presentation] Global existence and asymptotic behavior of solutions to a Hartree-type wave equation (2012)
  • Author(s): 津田谷公利
  • Organizer: 第29回九州における偏微分方程式研究集会
  • Place of Presentation: 九州大学(招待講演)
  • Year and Date: 2012-01-25
• [Presentation] Global existence of solutions of the wave equation of Hartree type (2011)
  • Author(s): 津田谷公利
  • Organizer: 応用数学セミナー
  • Place of Presentation: 東北大学(招待講演)
  • Year and Date: 2011-10-13
• [Presentation] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2011)
  • Author(s): K. Tsutaya
  • Organizer: Nonlinear Wave and Dispersive equations
  • Place of Presentation: 京都大学
  • Year and Date: 2011-02-15
• [Presentation] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2011)
  • Author(s): 津田谷公利
  • Organizer: Nonlinear Wave and Dispersive Equations
  • Place of Presentation: 京都大学(京都市)
  • Year and Date: 2011-02-15
• [Presentation] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2011)
  • Author(s): 津田谷公利
  • Organizer: 青葉山研究会
  • Place of Presentation: 東北大学(仙台市)
  • Year and Date: 2011-01-21
• [Presentation] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2010)
  • Author(s): 津田谷公利
  • Organizer: 応用解析研究会
  • Place of Presentation: 早稲田大学(東京都新宿区)
  • Year and Date: 2010-10-23
• [Presentation] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2010)
  • Author(s): M. Nakamura and K. Tsutaya
  • 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] Global existence and asymptotic behavior of solutions for the Dirac equation of Hartree type (2010)
  • Author(s): 津田谷公利
  • Organizer: 日本数学会
  • Place of Presentation: 名古屋大学(名古屋市)
  • Year and Date: 2010-09-24
• [Presentation] On the asymptotic behavior of solutions of the wave equation of Hartree type
  • Author(s): 津田谷公利
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学
• [Presentation] Global existence and asymptotic behavior of solutions to a Hartree-type wave equation
  • Author(s): 津田谷公利
  • Organizer: トリニティーカレッジ偏微分方程式セミナー
  • Place of Presentation: ダブリン，アイルランド