Parabolic concavity of solutions of parabolic equations

• Project/Area Number: 25610023
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Tohoku University
• Principal Investigator: Ishige Kazuhiro 東北大学, 理学(系)研究科(研究院), 教授 (90272020)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2014: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000); Fiscal Year 2013: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: 放物型方程式 / 冪凸 / 放物型凸 / 凸性 / ミンコフスキー和 / 熱方程式 / 弱連立非線形問題

Outline of Final Research Achievements

We introduced a new method for the study of parabolic concavity properties of solutions to parabolic boundary value problems and proved that a solutions of a nonlinear parabolic equation has parabolic power concavity properties. Furthermore, applying the arguments to parabolic concavity properties, we obtained parabolic power concavities of solutions to nonlinear parabolic systems and the relationship among the Minkowski additions of domains and the solutions of parabolic boundary value problems.

Research Products

• [Journal Article] A note on parabolic power concavity (2014)
  • Author(s): Kazuhiro Ishige and Paolo Salani
  • Journal Title: Kodai Mathematical Journal Volume: 37 Issue: 3 Pages: 668-679
  • DOI: 10.2996/kmj/1414674615
  • NAID: 130004705918
  • ISSN: 0386-5991, 1881-5472
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Parabolic power concavity and parabolic boundary value problems (2014)
  • Author(s): Kazuhiro Ishige and Paolo Salani
  • Journal Title: Mathematische Annalen Volume: 358 Issue: 3-4 Pages: 1091-1117
  • DOI: 10.1007/s00208-013-0991-5
  • Peer Reviewed
• [Presentation] Power concavity in weakly coupled elliptic and parabolic systems (2015)
  • Author(s): 石毛和弘
  • Organizer: パターン生成とダイナミクスの解構造の探求
  • Place of Presentation: 北海道大学
  • Year and Date: 2015-06-21
• [Presentation] Concavity properties of solutions for parabolic equations (2015)
  • Author(s): 石毛和弘
  • Organizer: Seminar on Qualitative Theory of Differential Equations
  • Place of Presentation: Comenius University, Slovakia
  • Year and Date: 2015-05-21
  • Int'l Joint Research / Invited
• [Presentation] Parabolic Minkowski convolutions of solutions to parabolic boundary value problems (2015)
  • Author(s): 石毛和弘
  • Organizer: 浜松偏微分方程式セミナー
  • Place of Presentation: 静岡大学
  • Year and Date: 2015-05-12
  • Invited
• [Presentation] Parabolic Minkowski convolutions of solutions to parabolic boundary value problems (2015)
  • Author(s): 石毛和弘
  • Organizer: 名古屋大学微分方程式セミナー
  • Place of Presentation: 名古屋大学
  • Year and Date: 2015-05-11
  • Invited
• [Funded Workshop] 4th Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's (2015)
  • Place of Presentation: Palinuro, Italy
  • Year and Date: 2015-05-25