Theory on the Analysis of Spot Solution Motion in Reaction-Diffusion Systems on Bounded Domains

2024 Fiscal Year Final Research Report

• Project/Area Number: 23K22400
• Project/Area Number (Other): 22H01129 (2022-2023)
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Multi-year Fund (2024); Single-year Grants (2022-2023)
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Josai University (2024); Hokkaido University (2022-2023)
• Principal Investigator: EI SHIN-ICHIRO 城西大学, 理学部, 教授 (30201362)
• Co-Investigator(Kenkyū-buntansha): 黒田 紘敏 北海道大学, 理学研究院, 准教授 (80635657)
• Project Period (FY): 2024-04-01 – 2025-03-31
• Keywords: 反応拡散系 / スポット解 / 進行波 / 相互作用

Outline of Final Research Achievements

For spot solutions appearing in multivariable reaction-diffusion mathematical models, we have established a general theoretical foundation to examine the influence of domain shape and boundary conditions on solution behavior. Additionally, for specific domains such as disk-shaped regions, regions with boundaries that are nearly straight, regions with obstacles, or metric graphs with Kirchhoff boundary conditions, we have rigorously derived concrete equations of motion from the abstract results indicated by the general theory. As an example, when spot solutions exhibit repulsive interactions, we have demonstrated that on metric graphs, motions such as approaching junction points or moving towards thicker paths can occur. Furthermore, we have successfully analyzed the motion of spot solutions based on the shape of obstacles.

Free Research Field

非線形解析

Academic Significance and Societal Importance of the Research Achievements

領域形状や境界条件がスポット解に与える影響は, これまで数値シミュレーションを中心に行われてきたが, 理論的に解析するための手法を与えることは, これまで経験的に行われてきた相互作用解析に確固たる数学的基盤を与え, なにが正しい答であるのかを明白にすることができるという点で学術的に意義がある．また, 応用範囲も広く, 実際に近い神経網における神経パルスの伝搬や障害物周りの散乱現象を正確に捉えるための理論的手法を与えることができることから, 数理科学のみならず, 生命系や工学系の問題への適用も期待され, 社会的な意義も十分に備えている．