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

• Project/Area Number: 20H01816
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Meiji 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
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥17,420,000 (Direct Cost: ¥13,400,000、Indirect Cost: ¥4,020,000); Fiscal Year 2023: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000); Fiscal Year 2022: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000); Fiscal Year 2021: ¥5,720,000 (Direct Cost: ¥4,400,000、Indirect Cost: ¥1,320,000); Fiscal Year 2020: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
• 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 / 特異極限法 / 伝播現象 / 全域解 / 特異極限系

Outline of Research at the Start

時間発展を伴うさまざまな現象は，非線形放物型偏微分方程式で記述されルことが多い．本研究課題では主に反応拡散系を取り挙げる．反応拡散系の解のダイナミクスは数値的にはいろいろと調べられているが，数学的には解のダイナミクスはあまり分かっ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，特異摂動系を用いる解析手法を提案し，全域解のもつ普遍的な数理構造を抽出することで，解のダイナミクスの決定を行う．

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

• [Int'l Joint Research] National Taiwan University/National Yang Ming Chiao Tung University(その他の国・地域)
• [Int'l Joint Research] Worcester Polytech Institute(米国)
• [Int'l Joint Research] National Taiwan University/National Yang Ming Chiao Tung University(その他の国・地域)
• [Int'l Joint Research] Worcester Polytech Institute(米国)
• [Int'l Joint Research] University of Aix-Marseille(フランス)
• [Int'l Joint Research] 国立台湾大学/国立交通大学(その他の国・地域)
• [Int'l Joint Research] Worcester Research Institute(米国)
• [Int'l Joint Research] National Taiwan University/National Tainan University/National Yang Ming Chiao Tung Univserity(その他の国・地域)
• [Int'l Joint Research] University of Wisconsin-Madison/Carnegie Mellon Universit/Worcester Polytech Institute(米国)
• [Int'l Joint Research] Oxford University(英国)
• [Journal Article] Example of Turing's instability by equal diffusion (2024)
  • Author(s): Ninomiya Hirokazu
  • Journal Title: Journal of Differential Equations Volume: 392 Pages: 255-265
  • DOI: 10.1016/j.jde.2024.02.026
  • Peer Reviewed
• [Journal Article] Weak entire solutions of reaction-interface systems (2023)
  • Author(s): Chen Yan-Yu, Ninomiya Hirokazu and Wu Chang-Hong
  • Journal Title: Discrete and Continuous Dynamical Systems - B Volume: 28 Issue: 12 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 equation (2023)
  • Author(s): Ito Shun and Ninomiya Hirokazu
  • Journal Title: Japan Journal of Industrial and Applied Mathematics Volume: 40 Issue: 2 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 states (2023)
  • Author(s): Iida Masato、Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai Nishi, Miyazaki, 889-2192, Japan、Izuhara Hirofumi、Kon Ryusuke
  • Journal Title: Discrete and Continuous Dynamical Systems - B Volume: 28 Issue: 12 Pages: 6159-6178
  • DOI: 10.3934/dcdsb.2023098
  • Peer Reviewed
• [Journal Article] On the rate of convergence in homogenization of time-fractional Hamilton-Jacobi equations (2023)
  • Author(s): H. Mitake, S. Sato,
  • Journal Title: NoDEA Nonlinear Differential Equations Appl. Volume: 30 Issue: 5 Pages: 30-68
  • DOI: 10.1007/s00030-023-00880-w
  • Peer Reviewed
• [Journal Article] On weak solutions to first-order discount mean field games (2023)
  • Author(s): H. Mitake, K. Terai
  • Journal Title: Minimax Theory and its Applications Volume: 8
  • Peer Reviewed
• [Journal Article] Spatial segregation of multiple species: A singular limit approach (2023)
  • Author(s): Izuhara Hirofumi, Monobe Harunori, Wu Chang-Hong
  • Journal Title: Discrete and Continuous Dynamical Systems - B Volume: 28 Issue: 12 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 Reaction (2023)
  • Author(s): Monobe Harunori、Shimojo Masahiko、Yanagida Eiji
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 55 Issue: 3 Pages: 2261-2287
  • DOI: 10.1137/22m1492544
  • Peer Reviewed
• [Journal Article] Traveling front solutions for perturbed reaction-diffusion equations (2023)
  • Author(s): Wah Wah and Masaharu Taniguchi
  • Journal Title: Mathematical Journal of Okayama University Volume: 65
  • Peer Reviewed
• [Journal Article] Quasiconvexity preserving property for fully nonlinear nonlocal parabolic equations (2022)
  • Author(s): Takashi Kagaya, Qing Liu, Hiroyoshi Mitake
  • Journal Title: Nonlinear Differential Equations and Applications NoDEA Volume: 30 Issue: 1 Pages: 1-28
  • DOI: 10.1007/s00030-022-00818-8
  • Peer Reviewed / Open Access
• [Journal Article] Global existence and uniqueness of solutions for one-dimensional reaction-interface systems (2022)
  • Author(s): Chen Yan-Yu、Ninomiya Hirokazu、Wu Chang-Hong
  • Journal Title: Journal of Differential Equations Volume: 324 Pages: 102-130
  • DOI: 10.1016/j.jde.2022.04.004
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Stationary solutions of an area-preserving curvature flow in an inhomogeneous medium (2022)
  • Author(s): Lui R.、Ninomiya H.
  • Journal Title: Proceedings of the American Mathematical Society Volume: 150 Pages: 2095-2105
  • DOI: 10.1090/proc/15787
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Global existence of solutions of area-preserving curvature flow of a convex plane curve in an inhomogeneous medium (2022)
  • Author(s): Lui R.、Ninomiya H.
  • Journal Title: Partial Differential Equations and Applications Volume: 3:42 Issue: 3 Pages: 1-16
  • DOI: 10.1007/s42985-022-00176-1
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Level-set forced mean curvature flow with the Neumann boundary condition (2022)
  • Author(s): J. Jang, D. Kwon, H. Mitake, H. V. Tran
  • Journal Title: J. Math. Pures Appl. Volume: 168 Pages: 143-167
  • DOI: 10.1016/j.matpur.2022.11.002
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The large time profile for Hamilton-Jacobi-Bellman equations (2022)
  • Author(s): D. A. Gomes, H. Mitake, H. V. Tran
  • Journal Title: Math. Ann. Volume: 384 Issue: 3-4 Pages: 1409-1459
  • DOI: 10.1007/s00208-021-02320-5
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation (2022)
  • Author(s): Giga Yoshikazu、Mitake Hiroyoshi、Sato Shoichi
  • Journal Title: Journal of Differential Equations Volume: 316 Pages: 364-386
  • DOI: 10.1016/j.jde.2022.01.057
  • NAID: 120007145351
  • Peer Reviewed / Open Access
• [Journal Article] Convergence to a traveling wave in the logarithmic diffusion equation with a bistable nonlinearity (2022)
  • Author(s): H. Matsuzawa, H. Monobe, M. Shimojo and E. Yanagida
  • Journal Title: Indiana University Mathematics Journal Volume: 71 Issue: 1 Pages: 125-151
  • DOI: 10.1512/iumj.2022.71.8850
  • Peer Reviewed
• [Journal Article] Entire solutions of the Allen?Cahn?Nagumo equation in a multi-dimensional space (2021)
  • Author(s): Ninomiya Hirokazu、,School of Interdisciplinary Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan
  • Journal Title: Discrete & Continuous Dynamical Systems - A Volume: 41 Issue: 1 Pages: 395-412
  • DOI: 10.3934/dcds.2020364
  • Peer Reviewed
• [Journal Article] Localized and Expanding Entire Solutions of Reaction?Diffusion Equations (2021)
  • Author(s): Hamel F.、Ninomiya H.
  • Journal Title: Journal of Dynamics and Differential Equations Volume: - Issue: 4 Pages: 2937-2974
  • DOI: 10.1007/s10884-020-09936-2
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Comparative analysis of continuum angiogenesis models (2021)
  • Author(s): Martinson W. Duncan、Ninomiya Hirokazu、Byrne Helen M.、Maini Philip K.
  • Journal Title: Journal of Mathematical Biology Volume: 82 Issue: 4 Pages: 1-34
  • DOI: 10.1007/s00285-021-01570-w
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Global Dynamics on One-Dimensional Excitable Media (2021)
  • Author(s): Chen Yan-Yu、Ninomiya Hirokazu、Wu Chang-Hong
  • Journal Title: SIAM Journal on Mathematical Analysis Volume: 53 Issue: 6 Pages: 7081-7112
  • DOI: 10.1137/20m1343014
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Existence of asymptotic speed of solutions to birth and spread type nonlinear partial differential equations (2021)
  • Author(s): Y. Giga, H. Mitake, T. Ohtsuka, H. V. Tran
  • Journal Title: Indiana Univ. Math. J. Volume: 70 Pages: 121-156
  • NAID: 120006501211
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model (2021)
  • Author(s): H. Mitake, H. V. Tran, T. S. Van
  • Journal Title: Commun. Math. Sci. Volume: 19 Pages: 495-512
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force (2021)
  • Author(s): H. Mitake, L. Zhang
  • Journal Title: Partial Differ. Equ. Appl. Volume: 2 Issue: 3 Pages: 1-24
  • DOI: 10.1007/s42985-020-00066-4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Compact traveling waves for anisotropic curvature flows with driving force (2021)
  • Author(s): Harunori Monobe, Hirokazu Ninomiya
  • Journal Title: Transactions of The American Mathematical Society Volume: 20 Issue: 4 Pages: 2447-2477
  • DOI: 10.1090/tran/8168
  • Peer Reviewed
• [Journal Article] A reaction-diffusion approximation of a semilinear wave equation (2021)
  • Author(s): Ninomiya Hirokazu、Yamamoto Hiroko
  • Journal Title: Journal of Differential Equations Volume: 272 Pages: 289-309
  • DOI: 10.1016/j.jde.2020.09.031
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The formation of spreading front : the singular limit of three-component reaction-diffusion models (2021)
  • Author(s): Hirofumi Izuhara, Harunori Monobe, Chang-Hong Wu,
  • Journal Title: Journal of Mathematical Biology Volume: 82 Issue: 5 Pages: 1-33
  • DOI: 10.1007/s00285-021-01591-5
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Bifurcation from infinity with applications to reaction-diffusion systems (2020)
  • Author(s): Chihiro Aida, Chao-Nien Chen, Kousuke Kuto, Hirokazu Ninomiya
  • Journal Title: Discrete and Continuous Dynamical Systems A Volume: 40 Issue: 6 Pages: 3031-3055
  • DOI: 10.3934/dcds.2020053
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] A level set approach for multi-layered interface systems (2020)
  • Author(s): Mitake Hiroyoshi、Ninomiya Hirokazu、Todoroki Kenta
  • Journal Title: Interfaces and Free Boundaries Volume: 22 Issue: 4 Pages: 383-400
  • DOI: 10.4171/ifb/444
  • Peer Reviewed / Int'l Joint Research
• [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: 62 Pages: 197-210
  • NAID: 120006778831
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Axisymmetric traveling fronts in balanced bistable reaction-diffusion equations (2020)
  • Author(s): Masaharu Taniguchi
  • Journal Title: Discrete & Continuous Dynamical Systems - A Volume: 40 Issue: 6 Pages: 3981-3995
  • DOI: 10.3934/dcds.2020126
  • NAID: 120006980496
  • Peer Reviewed
• [Journal Article] Traveling fronts in balanced bistable reaction-diffusion equations (2020)
  • Author(s): Masaharu Taniguchi
  • Journal Title: Advanced Studies in Pure Mathematics, The Role of Metrics in the Theory of Partial Differential Equations, Volume: 85 Pages: 417-428
  • Peer Reviewed
• [Presentation] Turing’s instability by equal diffusion (2024)
  • 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 media (2024)
  • Author(s): H. Ninomiya
  • Organizer: East Asia Workshop on Nonlinear Evolution Equations
  • Int'l Joint Research / Invited
• [Presentation] Turing’s instability by equal diffusion (2024)
  • 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 spirals (2024)
  • 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 diffusion (2023)
  • Author(s): Hirokazu Ninomiya
  • Organizer: 13th AIMS conference
  • Int'l Joint Research / Invited
• [Presentation] 等拡散によるパターン形成の例 (2023)
  • Author(s): 二宮広和
  • Organizer: 日本数学会
• [Presentation] 非一様場における面積保存曲率流のダイナミクス (2023)
  • Author(s): 二宮広和
  • Organizer: 日本数学会
• [Presentation] 面積保存曲率流のダイナミクス (2023)
  • Author(s): 二宮広和
  • Organizer: 北陸応用数理研究会2023
• [Presentation] Dynamics of area-preserving curvature flow in an inhomogeneous medium (2022)
  • Author(s): H. Ninomiya
  • Organizer: BIRS conference: Interfacial Phenomena in Reaction-Diffusion Systems (22w5165)
  • Int'l Joint Research / Invited
• [Presentation] Compact traveling waves for a mean-curvature flow with driving force (2022)
  • Author(s): Harunori Monobe
  • Organizer: A ReaDiNet seminar dar on reaction-diffusion system in biology
  • Int'l Joint Research / Invited
• [Presentation] Area-preserving curvature flow in an inhomogeneous medium (2021)
  • Author(s): H. Ninomiya
  • Organizer: An Online Conference on Recent Topics in Reaction-Diffusion System, Biology, Medicine and Chemistry
  • Int'l Joint Research / Invited
• [Presentation] On the equivalence of viscosity solutions and distributional solutions for the time-fractional diffusion equation (2021)
  • Author(s): H. Mitake
  • Organizer: The Hong Kong University of Science and Technology, PDE seminar
  • Int'l Joint Research / Invited
• [Presentation] On weak solutions to first-order discount mean field games (2021)
  • Author(s): H. Mitake
  • Organizer: OIST Analysis Seminar
  • Int'l Joint Research / Invited
• [Presentation] 時間分数冪拡散方程式に対する粘性解と超関数解の同値性 (2021)
  • Author(s): H. Mitake
  • Organizer: 京都大学NLPDEセミナー
  • Invited
• [Presentation] 凝結・分裂モデルに現れるハミルトン・ヤコビ方程式 (2021)
  • Author(s): H. Mitake
  • Organizer: 第 38 回 九 州における偏微分方程式研究集会
• [Presentation] Fisher-Stefan problems and the singular limit of reaction-diffusion systems (2021)
  • Author(s): Harunori Monobe
  • Organizer: Czech-Japanese Seminar in Applied Mathematics
  • Int'l Joint Research / Invited
• [Presentation] Large time behavior for a Hamilton-Jacobi equation in a critical Coagulation-Fragmentation model (2020)
  • Author(s): H. Mitake
  • Organizer: 2020 Seoul-Tokyo Conference
  • Int'l Joint Research / Invited
• [Presentation] 退化粘性 Hamilton-Jacobi 方程式の長時間挙動 (2020)
  • Author(s): H. Mitake
  • Organizer: 2020 年度「微分方程式の総合的研究」
  • Invited
• [Presentation] Fast reaction limit of a three-components Lotka-Volterra competition-diffusion system (2020)
  • Author(s): Harunori Monobe
  • Organizer: ``An online conference on mathematical biology'' ReaDiNet
  • Int'l Joint Research / Invited
• [Presentation] Fast reaction limit of three-components reaction-diffusion systems and free boundary problems describing population dynamics (2020)
  • Author(s): Harunori Monobe
  • Organizer: Interfacial Phenomena in Reaction-Diffusion Systems
  • Int'l Joint Research / Invited
• [Book] Traveling Front Solutions in Reaction-Diffusion Equations (2021)
  • Author(s): Masaharu Taniguchi
  • Total Pages: 170
  • Publisher: Mathematical Society of Japan
  • ISBN: 9784864970976
• [Remarks]
  • URL: http://www.isc.meiji.ac.jp/~nino38/index.html
• [Remarks]
  • URL: https://www.math.okayama-u.ac.jp/~taniguchi/publications/index.html
• [Funded Workshop] 14th Taiwan-Japan joint workshop for young scholars in applied mathematics (2024)
• [Funded Workshop] One-day workshop on PDEs and related topics (2024)
• [Funded Workshop] Geometric Aspects of Partial Differential Equations (2023)
• [Funded Workshop] Pattern dynamics appearing in mathematical biology (2023)
• [Funded Workshop] BIRS conference: Interfacial Phenomena in Reaction-Diffusion Systems (22w5165) (2022)
• [Funded Workshop] Modeling and Mathematical Analysis of Dynamics of Patterns (2021)
• [Funded Workshop] Interfacial Phenomena in Reaction-Diffusion Systems (2020)