An (N-2)-dimensional surface with positive principal curvatures gives an N-dimensional traveling front in bistable reaction-diffusion equations

• Project/Area Number: 26400169
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Okayama University
• Principal Investigator: Masaharu Taniguchi 岡山大学, 異分野基礎科学研究所, 教授 (30260623)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 進行波 / Allen--Cahn方程式 / 多次元 / 非対称 / 反応拡散方程式 / 多次元進行波 / Allen-Cahn方程式 / 国際研究者交流（Canada）

Outline of Final Research Achievements

In this project, I consider a parabolic equation with a bistable nonlinear term. This equation is called the Allen--Cahn equation or the Nagumo equation.The aim of this project is to search unknown traveling fronts. The result is as follows. For every given compact convex set in the (N-1)-Euclidean space, I proved the existence
of an N-dimensional traveling front solution associated with this set. Moreover, I proved that this traveling front solution is asymtotically stable if the given perturbation decays at infinity. These results were published by SIAM J. Math. Anal. 2015 and by J. Differential Equations 2016.

Academic Significance and Societal Importance of the Research Achievements

Allen--Cahn方程式(Nagumo方程式)は二層問題を記述するもっとも基本的な数理物理モデルである。形を保ったままで一定速度で伝播する波は進行波とよばれ，近年，多次元進行波の研究が求められている。この方程式の拡散項と反応項のいずれにも異方性すなわち方向依存性は入っておらず，空間的に等方的なモデルである。本研究では進行軸に対して軸非対称な多次元進行波の存在が始めて証明された。またこの軸非対称進行波が与えられた擾乱が無限遠方で減衰するとき，安定に伝播することが解明された。この事実は数理物理モデルにおける二層問題において情報がどのように伝達されるかを解明する手がかりになると期待される。

Research Products

• [Journal Article] Existence and stability of stationary solutions to the Allen--Cahn equation discretized in space and time (2020)
  • Author(s): Amy Poh Ai Ling and Masaharu Taniguchi
  • Journal Title: Mathematical Journal of Okayama University Volume: 未定
  • NAID: 120006778831
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Convex compact sets in $\mathbb{R}^{\scriptscriptstyle N-1}$ give traveling fronts of cooperation-diffusion systems in $\mathbb{R}^{\scriptscriptstyle N}$ (2016)
  • Author(s): Masaharu Taniguchi
  • Journal Title: Journal of Differential Equations Volume: 260 Issue: 5 Pages: 4301-4338
  • DOI: 10.1016/j.jde.2015.11.010
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen-Cahn equation (2015)
  • Author(s): Masaharu Taniguchi
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 47 Issue: 1 Pages: 455-476
  • DOI: 10.1137/130945041
  • NAID: 120005657667
  • Peer Reviewed / Open Access
• [Presentation] Multidimensional traveling fronts in reaction-diffusion equations (2018)
  • Author(s): Masaharu Taniguchi
  • Organizer: AIMS Conference on Dynamical Systems, Differential Equations and Applications, Taiwan
  • Int'l Joint Research / Invited
• [Presentation] Multidimensional traveling fronts in reaction-diffusion equations (2018)
  • Author(s): 谷口雅治
  • Organizer: The 11th MSJ-SI The Role of Metrics in the Theory of Partial Differential Equations
  • Int'l Joint Research / Invited
• [Presentation] Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations (2018)
  • Author(s): Masaharu Taniguchi
  • Organizer: Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type, MATRIX Research Centre, Australia
  • Invited
• [Presentation] Axially non-symmetric traveling fronts in balanced bistable reaction-diffusion equations (2018)
  • Author(s): 谷口雅治
  • Organizer: 日本数学会秋季総合分科会
• [Presentation] Allen--Cahn方程式における角錐型進行波の一意性と安定性 (2017)
  • Author(s): 谷口雅治
  • Organizer: 日本数学会秋季総合分科会 函数方程式分科会
• [Presentation] An (N-1)-dimensional convex compact set gives an N-dimensional traveling front (2017)
  • Author(s): Masaharu Taniguchi
  • Organizer: Equadiff2017
  • Int'l Joint Research / Invited
• [Presentation] An (N-1)-dimensional convex compact set gives an N-dimensional traveling front (2017)
  • Author(s): 谷口雅治
  • Organizer: 研究集会「非線形偏微分方程式の定性的理論」
  • Invited
• [Presentation] An (N-1)-dimensional convex compact set gives an N-dimensional traveling front (2016)
  • Author(s): Masaharu Taniguchi
  • Organizer: International Conference for the 70th Anniversary of Korean Mathematical Society
  • Place of Presentation: Seoul National University
  • Year and Date: 2016-10-20
  • Int'l Joint Research / Invited
• [Presentation] An (N-1)-dimensional convex compact set gives an N-dimensional traveling front (2016)
  • Author(s): Masaharu Taniguchi
  • Organizer: 11th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
  • Place of Presentation: Orlando, USA
  • Year and Date: 2016-07-01
  • Int'l Joint Research / Invited
• [Presentation] Multidimensional traveling fronts in reaction-diffusion equations (2015)
  • Author(s): Masaharu Taniguchi
  • Organizer: The Tenth East China Partial Differential Equations Conference
  • Place of Presentation: East China Normal University, 中華人民共和国
  • Year and Date: 2015-06-15
  • Invited
• [Presentation] (N-1)次元空間における狭義凸なコンパクト図形はN次元空間における競合拡散方程式系の進行波解を与える (2015)
  • Author(s): 谷口雅治
  • Organizer: 拡散に付随する数理科学セミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2015-05-30
  • Invited
• [Presentation] Convex compact sets in $\mathbb{R}^{N-1}$ give traveling fronts of cooperation-diffusion systems in $\mathbb{R}^{N}$ (2015)
  • Author(s): 谷口雅治
  • Organizer: 東工大数理解析研究会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2015-02-09
  • Invited
• [Presentation] An (N-1)-dimensional convex compact set gives an N-dimensional traveling front in the Allen-Cahn equation (2015)
  • Author(s): 谷口雅治
  • Organizer: FMSPミニワークショップ "Recent Trends in Traveling Waves"
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2015-01-30
  • Invited
• [Presentation] Convex compact sets in $\mathbb{R}^{N-1}$ give traveling fronts of cooperation-diffusion systems in $\mathbb{R}^{N}$ (2014)
  • Author(s): 谷口雅治
  • Organizer: 日本数学会秋季総合分科会函数方程式分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25 – 2014-09-28
• [Presentation] Convex compact sets in $\mathbb{R}^{N-1}$ give traveling fronts in $\mathbb{R}^{N}$ in cooperative diffusion systems (2014)
  • Author(s): Masaharu Taniguchi
  • Organizer: BIRS Workshop 14w5017
  • Place of Presentation: Banff International Research Station, Canada
  • Year and Date: 2014-05-25 – 2014-05-30
  • Invited
• [Funded Workshop] Okayama Workshop on Partial Differential Equations (2018)
• [Funded Workshop] パターン形成の数理とその周辺 (2018)
• [Funded Workshop] 研究集会「非線形偏微分方程式の非線形偏微分方程式の定性的理論」 (2017)
• [Funded Workshop] Okayama Workshop on Partial Differential Equations (2017)