Global stability of multi-dimensional traveling fronts

• Project/Area Number: 18540208
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: TANIGUCHI Masaharu Tokyo Institute of Technology, 大学院・情報理工学研究科, 准教授 (30260623)
• Project Period (FY): 2006 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥4,150,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥750,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 多次元進行波 / 安定性 / 反応拡散方程式 / 進行波 / 多次元 / 角錐型 / traveling wave / pyramidal shape / Allen-Cahn / 角錐型進行波 / Allen-Cahn方程式

Research Abstract

I studied the Allen-Cahn equation (Nagumo equation) in three-dimensional Euclidean space, and constructed pyramidal traveling front solutions and convex polyhedral traveling front solutions. I also proved that they are stable for given fluctuations.

Research Products

• [Journal Article] The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations (2009)
  • Author(s): M. Taniguchi
  • Journal Title: Journal of Differential Equations Vol.246 Pages: 2103-2130
  • 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 Vol.34,Issue9 Pages: 976-1002
  • Peer Reviewed
• [Journal Article] Pyramidal traveling fronts in the Allen-Cahn equations (2009)
  • Author(s): M. Taniguchi
  • Journal Title: 京都大学数理解析研究所講究録 1651 Pages: 1880-2818
• [Journal Article] The uniqueness and asymptotic stabilityi of pyramidal traveling fronts in the Allen-Cahn equations (2009)
  • Author(s): M.Taniguchi
  • Journal Title: Journal of Differential Equations 246
  • 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 34
  • Peer Reviewed
• [Journal Article] The uniqueness and asymptotic stability of pyramidal traveling fronts in the Allen-Cahn equations (2009)
  • Author(s): Masaharu Taniguchi
  • Journal Title: Journal of Differential Equations 246 Pages: 2103-2130
  • 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 (印刷中)
  • Peer Reviewed
• [Journal Article] Traveling fronts of pyramidal shapes in the Allen-Cahn equations (2007)
  • Author(s): M. Taniguchi
  • Journal Title: SIAM Journal on Mathematical Analysis Vol.39,No.1 Pages: 319-344
  • NAID: 120006581842
  • Peer Reviewed
• [Journal Article] The condition on the stability of stationary lines in a curvature flow in the whole plane (2007)
  • Author(s): M. Nara, M. Taniguchi
  • Journal Title: ournal of Differential Equations Vol.237,No.1 Pages: 61-76
  • Peer Reviewed
• [Journal Article] The condition on the stability of stationary lines in a curvature flow in the whole plane (2007)
  • Author(s): M. Nara and M. Taniguchi
  • Journal Title: Journal of Differential Equations Vol.237, No.1 Pages: 61-76
  • Peer Reviewed
• [Journal Article] Traveling fronts of pyramidal shapes in the Allen-Cahn equations (2007)
  • Author(s): M. Taniguchi
  • Journal Title: SIAM Journal on Mathematical Analysis Vol.39, No.1 Pages: 319-344
  • NAID: 120006581842
  • Peer Reviewed
• [Journal Article] Traveling fronts of pyramidal shapes in the Allen-Cahn equations (2007)
  • Author(s): Masaharu Taniguchi
  • Journal Title: SIAM Journal of Mathematical Analysis 39 Pages: 319-344
  • NAID: 120006581842
  • Peer Reviewed
• [Journal Article] The condition on the stability of stationary lines in a curvature flow in the whole plane (2007)
  • Author(s): M. Nara and M. Taniguchi
  • Journal Title: Journal of Differential Equations 237 Pages: 61-76
  • Peer Reviewed
• [Journal Article] Stability of a traveling wave in curvature flows for spatially non-decaying initial perturbations (2006)
  • Author(s): M. Nara, M. Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems Vol.14,No.1 Pages: 203-220
  • NAID: 120006581803
  • Peer Reviewed
• [Journal Article] Global stability of traveling curved fronts in the Allen-Cahn equations (2006)
  • Author(s): H. Ninomiya, M. Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems Vol.15,No.3 Pages: 819-832
  • NAID: 120006581802
  • Peer Reviewed
• [Journal Article] Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations (2006)
  • Author(s): M. Nara, M. Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems Vol16,No.1 Pages: 137-156
  • Peer Reviewed
• [Journal Article] Global stability of traveling curved fronts in the Allen-Cahn equations (2006)
  • Author(s): H. Ninomiya and M. Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems Vol.15, No.3 Pages: 819-832
  • NAID: 120006581802
  • Peer Reviewed
• [Journal Article] Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations (2006)
  • Author(s): M.Nara, M.Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems 16・1 Pages: 137-156
• [Journal Article] Global stability of traveling curved fronts in the Allen-Cahn equations (2006)
  • Author(s): H.Ninomiya, M.Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems 15・3 Pages: 819-832
  • NAID: 120006581802
• [Journal Article] Stability of traveling wave in curvature flows for spatially non-decaying initial perturbations (2006)
  • Author(s): M.Nara, M.Taniguchi
  • Journal Title: Discrete and Continuous Dynamical Systems 14・1 Pages: 203-220
  • NAID: 120006581803
• [Journal Article] Stability of planar waves in the Allen-Cahn equation
  • Author(s): H. Matano, M. Nara and M. Taniguchi
  • Journal Title: Communications in Partial Differential Equations (印刷中)
  • Peer Reviewed
• [Journal Article] The condition on the stability of stationary lines in a curvature Flow in the whole plane
  • Author(s): M.Nara, M.Taniguchi
  • Journal Title: Journal of Differential Equations (掲載決定)
• [Journal Article] Traveling fronts of pyramidal shapes in the Allen-Cahn equations
  • Author(s): M.Taniguchi
  • Journal Title: SIAM J. Math. Anal. (掲載決定)
  • NAID: 120006581842
• [Presentation] Pyramidal traveling fronts in multistable reaction-diffusion equations (2010)
  • Author(s): 谷口雅治
  • Organizer: 北東解析研究集会
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-02-22
• [Presentation] Multi-dimensional traveling fronts in bistable reaction-diffusion equations (2010)
  • Author(s): M. Taniguchi
  • Organizer: The 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Dresden University of Technology Dresden
  • Place of Presentation: Germany
• [Presentation] 多重安定な反応拡散方程式における角錐型進行波 (2009)
  • Author(s): 谷口雅治
  • Organizer: 数理解析セミナー
  • Place of Presentation: 首都大学東京
  • Year and Date: 2009-12-17
• [Presentation] Allen-Cahn方程式における角錐型進行波の一意性と安定性 (2008)
  • Author(s): 谷口雅治
  • Organizer: 日本数学会2008年度秋期総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] Allen-Cahn方程式における角錐型進行波の一意性と安定性 (2008)
  • Author(s): 谷口雅治
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] Traveling fronts of pyramidal shapes in the Allen-Cahn equations (2007)
  • Author(s): M. Taniguchi
  • Organizer: Minisymposium "Qualitative theory of elliptic and parabolic PDEs 1" Vienna, Austria EQUADIFF 2007
  • Place of Presentation: Vienna University of Technology
  • Year and Date: 2007-08-07
• [Presentation] Traveling fronts of pyramidal shapes in the Allen-Cahn equations (2007)
  • Author(s): Masaharu Taniguchi
  • Organizer: Equadiff2007
  • Place of Presentation: Vienna University of Technology
  • Year and Date: 2007-08-07
• [Presentation] Traveling fronts of pyramidal shapes in the Allen-Cahn equations (2007)
  • Organizer: Minisymposium "Qualitative theory of elliptic and parabolic PDEs 1" EQUADIFF 2007
  • Place of Presentation: Vienna University of Technology
• [Presentation] Traveling waves in the Allen-Cahn equations (2006)
  • Author(s): M. Taniguchi
  • Organizer: AIMS' Sixth International Conference on Dyn. Systems, Diff. Equations and Applications University of Poitiers
  • Place of Presentation: Poitiers
  • Year and Date: 2006-06-27
• [Presentation] Allen-Cahn方程式における多次元進行波 (2006)
  • Author(s): 谷口雅治
  • Organizer: 日本数学会2006年年会
  • Place of Presentation: 中央大学
  • Year and Date: 2006-03-28
• [Book] 現代数理科学辞典(第2版) (2010)
  • Author(s): 谷口雅治
  • Publisher: 丸善株式会社
• [Book] 現代数理科学事典第2版,XI-9-2 分担執筆(Page 1283-1289) (2010)
  • Author(s): 谷口雅治
  • Total Pages: 7
  • Publisher: 丸善株式会社
• [Book] 数学の言葉と論理 (2008)
  • Author(s): 渡辺治,北野晃朗,木村泰紀,谷口雅治
  • Total Pages: 219
  • Publisher: 朝倉書店
• [Book] 数学の言葉と論理 (2008)
  • Author(s): 渡辺治, 北野晃朗, 木村泰紀, 谷口雅治(著)
  • Publisher: 朝倉書店
• [Book] 数学の言葉と論理 (2008)
  • Author(s): 渡辺治, 北野晃朗, 木村泰紀, 谷口雅治
  • Total Pages: 219
  • Publisher: 朝倉書店
• [Remarks] ホームページ等