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2023 Fiscal Year Annual Research Report

Mathematical analysis of pattern dynamics of reaction-diffusion systems and their singular limit problems

Research Project

Project/Area Number 20H01816
Research InstitutionMeiji University

Principal Investigator

二宮 広和  明治大学, 総合数理学部, 専任教授 (90251610)

Co-Investigator(Kenkyū-buntansha) 飯田 雅人  宮崎大学, 工学部, 教授 (00242264)
谷口 雅治  岡山大学, 異分野基礎科学研究所, 教授 (30260623)
物部 治徳  大阪公立大学, 大学院理学研究科, 准教授 (20635809)
三竹 大寿  東京大学, 大学院数理科学研究科, 准教授 (90631979)
Project Period (FY) 2020-04-01 – 2024-03-31
Keywords反応拡散系 / 自由境界問題 / パターンダイナミクス / 進行波解 / 特異極限問題
Outline of Annual Research Achievements

非線形放物型偏微分方程式の解のダイナミクスを決定することは,非線形放物型偏微分方程式の理論的研究における重要な問題のひとつである.しかし,比較的簡単と考えられる反応拡散系でさえ,解のダイナミクスを決定できていないのが現状である.本研究課題では,反応拡散系の解のダイナミクスを決定するための解析手法の開発と普遍的な数理構造の抽出を行った.テーマ1としては,1次元単独反応拡散系の進行波解の速度の特徴づけを行った.また,この知見をもとに,多数の平衡点をもつ進行波解の特徴付けや単調でない進行波解の特徴付けを行う.複数成分の反応拡散系では,一般に比較定理が成り立たず,リャプノフ関数も存在しないので,ダイナミクスを決定するには,特殊な状況を考察する必要がある.そのため,テーマ2では,反応拡散系に小さなパラメータを適切に導入することで現れる遷移層の運動を記述する自由境界問題を調べる.多くの自由境界問題において,領域のパターンダイナミクスの情報を得ることは非常に難しい.従って,特異極限系の解の挙動の情報が得られるように反応拡散系に小さなパラメータを適切に導入する必要がある.研究代表者等が提案している「反応界面系」は,空間をパラメータとする無限連立常微分方程式とこの未知変数を外力に含む平均曲率流よりなる系である.1次元空間の反応界面系の解の大域挙動は3種類に分類できることをすでに示した.本年度は,1次元全域解の特徴付けを行った.また,多次元空間の反応界面系の全域解特徴付けの準備として,面積保存平均曲率流の大域解の特徴付けを行った.テーマ3では,面積保存曲率流の定常解や解の大域的存在についての結果を得た.

Research Progress Status

令和5年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和5年度が最終年度であるため、記入しない。

  • Research Products

    (21 results)

All 2024 2023 Other

All Int'l Joint Research (2 results) Journal Article (10 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 9 results) Presentation (5 results) (of which Int'l Joint Research: 5 results,  Invited: 5 results) Funded Workshop (4 results)

  • [Int'l Joint Research] National Taiwan University/National Yang Ming Chiao Tung University(その他の国・地域)

    • Country Name
      その他の国・地域
    • Counterpart Institution
      National Taiwan University/National Yang Ming Chiao Tung University
  • [Int'l Joint Research] Worcester Polytech Institute(米国)

    • Country Name
      U.S.A.
    • Counterpart Institution
      Worcester Polytech Institute
  • [Journal Article] Example of Turing's instability by equal diffusion2024

    • Author(s)
      Ninomiya Hirokazu
    • Journal Title

      Journal of Differential Equations

      Volume: 392 Pages: 255~265

    • DOI

      10.1016/j.jde.2024.02.026

  • [Journal Article] Weak entire solutions of reaction-interface systems2023

    • Author(s)
      Chen Yan-Yu、Ninomiya Hirokazu、Wu Chang-Hong
    • Journal Title

      Discrete and Continuous Dynamical Systems - B

      Volume: 28 Pages: 6015~6033

    • DOI

      10.3934/dcdsb.2022174

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Spatial homogenization by perturbation on the complex Ginzburg-Landau equation2023

    • Author(s)
      Ito Shun、Ninomiya Hirokazu
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics

      Volume: 40 Pages: 823~841

    • DOI

      10.1007/s13160-022-00556-5

    • Peer Reviewed
  • [Journal Article] Cross-diffusion predator?prey model derived from the dichotomy between two behavioral predator states2023

    • Author(s)
      Iida Masato、Izuhara Hirofumi、Kon Ryusuke
    • Journal Title

      Discrete and Continuous Dynamical Systems - B

      Volume: 28 Pages: 6159~6178

    • DOI

      10.3934/dcdsb.2023098

    • Peer Reviewed
  • [Journal Article] On the rate of convergence in homogenization of time-fractional Hamilton-Jacobi equations2023

    • Author(s)
      Mitake Hiroyoshi、Sato Shoichi
    • Journal Title

      Nonlinear Differential Equations and Applications NoDEA

      Volume: 30 Pages: 30-68

    • DOI

      10.1007/s00030-023-00880-w

    • Peer Reviewed
  • [Journal Article] Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations2023

    • Author(s)
      Kagaya Takashi、Liu Qing、Mitake Hiroyoshi
    • Journal Title

      Nonlinear Differential Equations and Applications NoDEA

      Volume: 30:13 Pages: 1-28

    • DOI

      10.1007/s00030-022-00818-8

    • Peer Reviewed
  • [Journal Article] On weak solutions to first-order discount mean field games2023

    • Author(s)
      H. Mitake, K. Terai
    • Journal Title

      Minimax Theory and its Applications

      Volume: 8 Pages: 139--170

    • Peer Reviewed
  • [Journal Article] Spatial segregation of multiple species: A singular limit approach2023

    • Author(s)
      Izuhara Hirofumi、Monobe Harunori、Wu Chang-Hong
    • Journal Title

      Discrete and Continuous Dynamical Systems - B

      Volume: 28 Pages: 6208~6232

    • DOI

      10.3934/dcdsb.2022215

    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Spreading and Extinction of Solutions to the Logarithmic Diffusion Equation with a Logistic Reaction2023

    • Author(s)
      Monobe Harunori、Shimojo Masahiko、Yanagida Eiji
    • Journal Title

      SIAM Journal on Mathematical Analysis

      Volume: 55 Pages: 2261~2287

    • DOI

      10.1137/22M1492544

    • Peer Reviewed
  • [Journal Article] Traveling front solutions for perturbed reaction-diffusion equations2023

    • Author(s)
      Wah Wah and Masaharu Taniguchi
    • Journal Title

      Mathematical Journal of Okayama University

      Volume: 65 Pages: 125--143

    • Peer Reviewed
  • [Presentation] Turing’s instability by equal diffusion2024

    • Author(s)
      Hirokazu Ninomiya
    • Organizer
      MATRIX-RIMS Tandem Workshop: Evolutionary Partial Differential Equations and Applications
    • Int'l Joint Research / Invited
  • [Presentation] Dynamics of area-preserving curvature flow in heterogeneous media2024

    • Author(s)
      H. Ninomiya
    • Organizer
      East Asia Workshop on Nonlinear Evolution Equations
    • Int'l Joint Research / Invited
  • [Presentation] Turing’s instability by equal diffusion2024

    • Author(s)
      H. Ninomiya
    • Organizer
      Turing Symposium on Morphogenesis
    • Int'l Joint Research / Invited
  • [Presentation] On asymptotic growth rate of solutions to level-set forced mean curvature flows with evolving spirals2024

    • Author(s)
      H. Mitake
    • Organizer
      East Asia Workshop on Nonlinear Evolution Equations
    • Int'l Joint Research / Invited
  • [Presentation] Example of Turing’s instability by equal diffusion2023

    • Author(s)
      Hirokazu Ninomiya
    • Organizer
      13th AIMS conference
    • Int'l Joint Research / Invited
  • [Funded Workshop] 14th Taiwan-Japan joint workshop for young scholars in applied mathematics2024

  • [Funded Workshop] One-day workshop on PDEs and related topics2024

  • [Funded Workshop] Geometric Aspects of Partial Differential Equations2023

  • [Funded Workshop] Pattern dynamics appearing in mathematical biology2023

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Published: 2024-12-25  

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