2023 Fiscal Year Final Research Report
Mathematical analysis of pattern dynamics of reaction-diffusion systems and their singular limit problems
| Project/Area Number |
20H01816
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Meiji University |
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Principal Investigator |
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| 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)
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| 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 |
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| Outline of Final Research Achievements |
In this research project, we developed an analysis method to determine the dynamics of solutions in reaction-diffusion systems and extracted universal mathematical structures. For single-component reaction-diffusion equations, we characterized the velocity of traveling wave solutions in a one-dimensional space and constructed entire solutions. For multiple-component reaction-diffusion systems, we investigated the dynamics of solutions to singular limit problems. Introducing the reaction interface system as a singular limit problem capable of handling dynamics, we proved that the global behavior of solutions to reaction interface systems in one-dimensional space can be classified into three types. Additionally, as a preparation for characterizing pattern dynamics of reaction-diffusion systems in multi-dimensional non-uniform media, we studied area-preserving mean curvature flow and successfully examined information on stationary solutions and dynamics under certain conditions.
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| Free Research Field |
非線形偏微分方程式
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| Academic Significance and Societal Importance of the Research Achievements |
多くの現象が，非線形偏微分方程式で記述されるが，その解の挙動は，数値計算を行わないと分からない場合がほとんどである．この研究課題では，解のダイナミクスを調べるために，解のダイナミクスがわかる新しい特異極限問題を導入した．また，単独反応拡散方程式の全域解の性質を抽出する手法を開発した．こうした手法の開発を重ねることで，将来的に非線形偏微分方程式の解挙動を表現する数学的言語が確立される．
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