Study on interfacial phenomena in nonlinear partial differential equations

• Project/Area Number: 21740113
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Iwate University (2011); The University of Tokyo (2009-2010)
• Principal Investigator: NARA Mitsunori 岩手大学, 人文社会科学部, 准教授 (90512161)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2011: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2010: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2009: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 関数方程式論 / 偏微分方程式 / 界面現象 / パターン形成

Research Abstract

In this study, we analyzed the asymptotic stability of planar waves and the large time behavior of disturbed planar fronts that appear in a reaction diffusion equation with bistable type nonlinearity. We improved some previous results and showed the approximation theorem for the large time behavior of planar fronts based on the mean curvature flow. The motion of hypersurfaces governed by the mean curvature flow was also studied, and some sufficient conditions for the asymptotic stability of hyperplanes as traveling wave solutions were shown.

Research Products

• [Journal Article] Large time behavior of disturbed planar fronts in the Allen-Cahn equation (2011)
  • Author(s): H. Matano, M. Nara
  • Journal Title: Journal of Differential Equations Volume: Vol.251 Pages: 3522-3557
  • Peer Reviewed
• [Journal Article] Large time behavior of disturbed planar fronts in the Allen-Cahn equation (2011)
  • Author(s): 俣野博, 奈良光紀
  • Journal Title: Journal of Differential Equations Volume: 251 Pages: 3522-3557
  • Peer Reviewed
• [Journal Article] Stability of planar waves in the Allen-Cahn equation (2009)
  • Author(s): H. Matano, M. Nara, M. Taniguchi
  • Journal Title: Communications in Partial Differential Equations Volume: Vol.34 Pages: 976-1002
  • Peer Reviewed
• [Journal Article] Stability of planar waves in the Allen-Cahn equation (2009)
  • Author(s): 俣野博, 奈良光紀, 谷口雅治
  • Journal Title: Communications in Partial Differential Equations Volume: 34 Pages: 976-1002
  • Peer Reviewed
• [Presentation] Allen-Cahn方程式における解の時間漸近挙動と平面波の安定性 (2012)
  • Author(s): 奈良光紀
  • Organizer: 平成23年度日本数学会 東北支部会
  • Place of Presentation: 岩手大学(岩手県)(招待講演)
  • Year and Date: 2012-02-18
• [Presentation] Allen-Cahn方程式における解の時間漸近挙動と平面波の安定性 (2012)
  • Author(s): 奈良光紀
  • Organizer: 平成23年度日本数学会東北支部会
  • Place of Presentation: 岩手大学
• [Presentation] Allen-Cahn方程式における解の時間漸近挙動と平面波の安定性 (2011)
  • Author(s): 奈良光紀
  • Organizer: 研究集会「非線形現象の数値シミュレーションと解析2011」
  • Place of Presentation: 北海道大学
• [Presentation] Large time behavior of disturbed planar fronts in the Allen-Cahn equation (2010)
  • Author(s): 奈良光紀
  • Organizer: The 8th AIMS Conference on Dynamical Systems
  • Place of Presentation: ドレスデン工科大学(ドイツ)
  • Year and Date: 2010-05-27
• [Presentation] Large time behavior of disturbed planar fronts in the Allen-Cahn equation (2010)
  • Author(s): 奈良光紀
  • Organizer: The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Dresden University of Technology (ドイツ)
• [Presentation] Allen-Cahn方程式と平均曲率流における進行波 (2009)
  • Author(s): 奈良光紀
  • Organizer: 日本数学会2009年度秋季総合分科会 応用数学分科会
  • Place of Presentation: 大阪大学
  • Year and Date: 2009-09-27
• [Presentation] Allen-Cahn方程式と平均曲率流における進行波 (2009)
  • Author(s): 奈良光紀
  • Organizer: 日本数学会2009年度秋季総合分科会応用数学分科会
  • Place of Presentation: 大阪大学