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Relation between pattern formations and complex singularities of solutions of nonlinear partial differential equations

Research Project

Project/Area Number 16K13778
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionMeiji University

Principal Investigator

Ninomiya Hirokazu  明治大学, 総合数理学部, 専任教授 (90251610)

Co-Investigator(Kenkyū-buntansha) 桂田 祐史  明治大学, 総合数理学部, 専任准教授 (80224484)
池田 幸太  明治大学, 総合数理学部, 専任准教授 (50553369)
小野寺 有紹  東京工業大学, 理学院, 准教授 (70614999)
Project Period (FY) 2016-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2017: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2016: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywords反応拡散系 / 複素特異点 / パターン形成 / 自由境界問題 / 複素領域 / パターンダイナミクス / 爆発問題 / 全域解 / ヘレ・ショウ問題 / 複素数特異点 / 非線形偏微分方程式
Outline of Final Research Achievements

We showed the analytic extension of the solution of a reaction-diffusion equation into complex domain. However, in general, the complex singularities appear. To study the motion of the singularities, we consider the heat equation and the Allen-Cahn-Nagumo equation. For the Allen-Cahn-Nagumo equation, we study the motion of the complex singularities of the exact solutions. We tried to extend the properties derived from this observation into the case for general nonlinearity.
Moreover, we study the bifurcation from infinity and we get the result which generalizes the results of Stuart and Rabinowitz into the multi-component reaction-diffusion systems.

Academic Significance and Societal Importance of the Research Achievements

非線形偏微分方程式は,天気予報などの身近な問題だけでなく,ナノテクノロジーから宇宙の解明まで幅広い分野で利用されている.しかし,非線形偏微分方程式の解を表現する解の公式はないため,解の形状を表現する手法の開発が求められている.本研究課題では,複素特異点や無限遠からのパターン形成について考察し,一部は論文として投稿するに至った.

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (58 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (5 results) Journal Article (16 results) (of which Int'l Joint Research: 4 results,  Peer Reviewed: 16 results,  Acknowledgement Compliant: 4 results,  Open Access: 1 results) Presentation (37 results) (of which Int'l Joint Research: 26 results,  Invited: 27 results)

  • [Int'l Joint Research] 国立台湾大学/国立台南大学(その他の国・地域)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Minnesota/Worcester Polytech Institute(米国)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] 国立台湾大学/淡江大学/国立台南大学(台湾)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] National Taiwan University/National Tainan University/Tamkang University(台湾)

    • Related Report
      2016 Research-status Report
  • [Int'l Joint Research] フィレンツェ大学(イタリア)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Traveling wave solutions for a bacteria system with density-suppressed motility2019

    • Author(s)
      Lui, R. and Ninomiya, H.
    • Journal Title

      Discrete & Continuous Dynamical Systems - B

      Volume: 24 Issue: 2 Pages: 931-940

    • DOI

      10.3934/dcdsb.2018213

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A review on reaction?diffusion approximation2018

    • Author(s)
      Iida M., Ninomiya H. and Yamamoto H.
    • Journal Title

      Journal of Elliptic and Parabolic Equations

      Volume: 4 Issue: 2 Pages: 565-600

    • DOI

      10.1007/s41808-018-0029-y

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Reaction diffusion approximation of nonlocal interactions using Jacobi polynomials2018

    • Author(s)
      Ninomiya Hirokazu、Tanaka Yoshitaro、Yamamoto Hiroko
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 印刷中 Issue: 2 Pages: 613-651

    • DOI

      10.1007/s13160-017-0299-z

    • NAID

      210000170293

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Entire solutions originating from monotone fronts to the Allen-Cahn equation2018

    • Author(s)
      Y.-Y. Chen, J.-S. Guo, H. Ninomiya and C.-H. Yao
    • Journal Title

      Physica D

      Volume: - Pages: 1-19

    • DOI

      10.1016/j.physd.2018.04.003

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Another interpretation of the Goldman-Hodgkin-Katz equation based on Ling's adsorption theory2018

    • Author(s)
      Tamagawa, H. and Ikeda, K.
    • Journal Title

      European Biophysics Journal

      Volume: 47 Issue: 8 Pages: 869-879

    • DOI

      10.1007/s00249-018-1332-0

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Traveling wave solutions with convex domains for a free boundary problem2017

    • Author(s)
      H. Monobe and H. Ninomiya
    • Journal Title

      Discrete and Continuous Dynamical Systems. Ser. A

      Volume: Volume 37, Issue 2 Issue: 2 Pages: 905-914

    • DOI

      10.3934/dcds.2017037

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Existence of rotating spots with spatially dependent feedback in the plane in a wave front interaction model2017

    • Author(s)
      Y.-Y. Chen, J.-S. Guo and H. Ninomiya
    • Journal Title

      Journal of Dynamics and Differential Equations

      Volume: 29 Issue: 2 Pages: 465-483

    • DOI

      10.1007/s10884-016-9541-7

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Traveling curved waves in two dimensional excitable media2017

    • Author(s)
      H. Ninomiya and C.-H. Wu
    • Journal Title

      SIAM J. Mathematical Analysis

      Volume: 49 Issue: 2 Pages: 777-817

    • DOI

      10.1137/16m1064040

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Reaction, diffusion and non-local interaction2017

    • Author(s)
      H. Ninomiya, Y. Tanaka and H. Yamamoto
    • Journal Title

      Journal of Mathematical Biology

      Volume: 75 Issue: 5 Pages: 1203-1233

    • DOI

      10.1007/s00285-017-1113-x

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Vanishing, moving and immovable interfaces in fast reaction limits2017

    • Author(s)
      M. Iida, H. Monobe, H. Murakawa and H. Ninomiya
    • Journal Title

      Journal of Differential Equations

      Volume: 印刷中 Issue: 5 Pages: 2715-2735

    • DOI

      10.1016/j.jde.2017.04.009

    • Related Report
      2017 Research-status Report 2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Mathematical approach to nonlocal interactions using a reaction-diffusion system2017

    • Author(s)
      Y. Tanaka, H. Yamamoto, and H. Ninomiya
    • Journal Title

      Development, Growth & Differentiation

      Volume: - Issue: 5 Pages: 388-395

    • DOI

      10.1111/dgd.12354

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Bifurcation structure of a car-following model with nonlinear dependence on the relative velocity2017

    • Author(s)
      A. Tomoeda, T. Miyaji, Tomoyuki, K. Ikeda
    • Journal Title

      Transportmetrica A: Transport Science

      Volume: 14 Issue: 5-6 Pages: 503-519

    • DOI

      10.1080/23249935.2017.1351007

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Generation of membrane potential beyond the conceptual range of Donnan theory and Goldman-Hodgkin-Katz equation2017

    • Author(s)
      H. Tamagawa, K.Ikeda
    • Journal Title

      Journal of Biological Physics

      Volume: 43 Issue: 3 Pages: 319-340

    • DOI

      10.1007/s10867-017-9454-7

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Remarks on the large time behavior of the L^2-norm of solutions to strongly damped wave equations2017

    • Author(s)
      Ryo Ikehata and Michiaki Onodera
    • Journal Title

      Differential Integral Equations

      Volume: 30 Pages: 505-520

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] A reaction diffusion model for understanding phyllotactic formation2016

    • Author(s)
      Y. Tanaka, M. Mimura and H. Ninomiya
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 33 Issue: 1 Pages: 183-205

    • DOI

      10.1007/s13160-015-0202-8

    • NAID

      210000175348

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Dynamical approach to an overdetermined problem in potential theory2016

    • Author(s)
      Michiaki Onodera
    • Journal Title

      J. Math. Pures Appl.

      Volume: 未定 Issue: 4 Pages: 768-796

    • DOI

      10.1016/j.matpur.2016.03.011

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] Propagation phenomena in reaction-diffusion equations2018

    • Author(s)
      H. Ninomiya
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Dynamics of interface-reaction systems2018

    • Author(s)
      H. Ninomiya
    • Organizer
      UK-Japan Workshop on Analysis of Nonlinear Differential Equations
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 単一自己駆動粒子モデルにおける準周期的な運動と振動運動の関係性2018

    • Author(s)
      池田幸太, 宮路 智行, 北畑 裕之, 小谷野 由紀, 義永 那津人
    • Organizer
      第28回非線形反応と協同現象研究会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Reduction approach to a reaction-diffusion system for collective motions of camphor boats2018

    • Author(s)
      Kota Ikeda, Shin-Ichiro Ei, Masaharu Nagayama, Akiyasu Tomoeda
    • Organizer
      Seminar on Sorbonne Universite
    • Related Report
      2018 Annual Research Report
  • [Presentation] デルタ関数を伴う反応拡散モデルにおける中心多様体縮約理論の構築2018

    • Author(s)
      池田 幸太, 栄 伸一郎, 友枝 明保, 長山 雅晴
    • Organizer
      第一回はこだて数理解析研究集会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 単一自己駆動粒子が呈する準周期的な運動の数理解析2018

    • Author(s)
      池田 幸太, 宮路 智行, 北畑 裕之, 小谷野 由紀, 義永 那津人
    • Organizer
      日本応用数理学会2018年度年会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 単一自己駆動粒子が呈する準周期的な運動の数理解析2018

    • Author(s)
      池田幸太, 宮路 智行, 北畑 裕之, 小谷野 由紀, 義永 那津人
    • Organizer
      札幌非線形現象研究会2018
    • Related Report
      2018 Annual Research Report
  • [Presentation] Reduction approach to a reaction-diffusion system for collective motions of camphor boats2018

    • Author(s)
      Kota Ikeda, Shin-Ichiro Ei, Masaharu Nagayama, Akiyasu Tomoeda
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Quasi-Periodic Solution in a Dynamical System for the Motion of a Single Particle2018

    • Author(s)
      Kota Ikeda, Hiroyuki Kitahata, Yuki Koyano, Tomoyuki Miyaji, Natsuhiko Yoshinaga
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Quasi-periodic solution in a dynamical system for the motion of a single particle2018

    • Author(s)
      Kota Ikeda, Hiroyuki Kitahata, Yuki Koyano, Tomoyuki Miyaji, Natsuhiko Yoshinaga
    • Organizer
      International Conference on the Dynamics of Differential Equations
    • Related Report
      2018 Annual Research Report
  • [Presentation] Reduction approach to a reaction-diffusion system for collective motions of camphor boats2018

    • Author(s)
      Kota Ikeda, Shin-Ichiro Ei, Masaharu Nagayama, Akiyasu Tomoeda
    • Organizer
      HMAセミナー
    • Related Report
      2018 Annual Research Report
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2018

    • Author(s)
      Michiaki Onodera
    • Organizer
      Workshop on Nonlinear Parabolic PDEs
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2018

    • Author(s)
      Michiaki Onodera
    • Organizer
      Geometric PDEs in Freiburg 2018
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Dynamical approach to an overdetermined problem2018

    • Author(s)
      Michiaki Onodera
    • Organizer
      Tokyo Tech -- Uppsala University 5th Joint Symposium
    • Related Report
      2018 Annual Research Report
  • [Presentation] Dynamics of solutions of interface-reaction systems2018

    • Author(s)
      H. Ninomiya
    • Organizer
      The 3rd International Conference on Dynamics of Differential Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Dynamics of interface-reaction equations2018

    • Author(s)
      H. Ninomiya
    • Organizer
      International Conference on Nonlinear Analysis and its Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 反応拡散方程式の進行波解と全域解2018

    • Author(s)
      二宮広和
    • Organizer
      日本数学会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Quasi-periodic solution in a dynamical system for the motion of a single particle2018

    • Author(s)
      池田幸太
    • Organizer
      International Conference on the Dynamics of Differential Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Traveling waves of curvature flows with anisotropic external forces2017

    • Author(s)
      H. Ninomiya
    • Organizer
      Mathematical Analysis on Nonlinear PDEs
    • Place of Presentation
      東北大学,仙台
    • Year and Date
      2017-01-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Reaction-diffusion approximations of non-local evolutionary equation and wave equation2017

    • Author(s)
      H. Ninomiya
    • Organizer
      International Conference on Elliptic and Parabolic Problems
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Dynamics of waves in excitable media2017

    • Author(s)
      H. Ninomiya
    • Organizer
      日本数理生物学会
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Dynamics of waves in excitable media2017

    • Author(s)
      H. Ninomiya
    • Organizer
      The 3rd conference on Differential Geometry and Differential equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Patterns and complex singularities in reaction-diffusion equations2017

    • Author(s)
      H. Ninomiya
    • Organizer
      PDE seminar, University of Minnesota
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Bernoulliの自由境界問題の極限解について2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      水戸数学・情報数理研究会2017
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      Ito Workshop on Partial Differential Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Flow approach to Bernoulli's free boundary problem2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      第64回幾何学シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Dynamical approach to an overdetermined problem2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      Geometry Seminar
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      The third Japanese-Spanish workshop on Differential Geometry
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      Uppsala University - Tokyo Tech Joint Symposium
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Hyperbolic solutions to Bernoulli's free boundary problem2017

    • Author(s)
      Michiaki Onodera
    • Organizer
      Seminaire d'Equations aux Derivees Partielles et Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the solution structure of Bernoulli's free boundary problem2017

    • Author(s)
      M. Onodera
    • Organizer
      The 9th Nagoya Workshop on Differential Equations
    • Place of Presentation
      名古屋大学,名古屋
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Obstacle-induced spiral in two-dimensional excitable media2016

    • Author(s)
      H. Ninomiya
    • Organizer
      ミクロな振る舞いと集団的パターン形成に係る階層的構造の解明
    • Place of Presentation
      数理解析研究所,京都
    • Year and Date
      2016-09-12
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Pattern formation through non-local interactions,2016

    • Author(s)
      H. Ninomiya
    • Organizer
      The 2016 (26th) annual meeting of the Japanese Society for Mathematical Biology
    • Place of Presentation
      九州大学,福岡
    • Year and Date
      2016-09-07
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Layered Interface Systems and its Dynamics2016

    • Author(s)
      H. Ninomiya
    • Organizer
      11th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Orlando USA
    • Year and Date
      2016-07-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Reduction Approach to a Reaction-diffusion System for Collective Motions of Camphor Boats2016

    • Author(s)
      K. Ikeda, S.-I. Ei, A. Tomoeda, M. Nagayama
    • Organizer
      11th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Orlando, Orlando, FL, USA
    • Year and Date
      2016-07-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Reduction approach to a reaction-diffusion system for collective motions of camphor boats2016

    • Author(s)
      K. Ikeda, S.-I. Ei, A. Tomoeda, M. Nagayama
    • Organizer
      Mathematics of Pattern Formation
    • Place of Presentation
      Mathematical Research and Conference Center, Bedlewo, Poland
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Dynamical approach to an overdetermined problem2016

    • Author(s)
      M. Onodera
    • Organizer
      Korea-Japan International Workshop of Nonlinear Partial Differential Equations --- Aspect of Regularity and Asymptotics ---
    • Place of Presentation
      洞爺サンパレス,北海道
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2016-04-21   Modified: 2022-02-22  

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