Structure of stationary solutions and motion of interfaces in bistable reaction-diffusion equations

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17569
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Osaka Prefecture University (2018); Tokyo Institute of Technology (2015-2017)
• Principal Investigator: Kan Toru 大阪府立大学, 理学(系)研究科(研究院), 准教授 (60647270)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 双安定反応拡散方程式 / 領域の特異極限 / 定常問題 / 分岐解析 / 安定性解析

Outline of Final Research Achievements

For reaction-diffusion equations, structure of stationary solutions and motion of interfaces of solutions were studied. I considered a dumbbell-shaped domain which converges to a one-dimensional interval and derived the limiting equation on the interval. Based on the analysis of stationary solutions of the limiting equation, I found stationary solutions of the equation on the dumbbell-shaped domain. In addition, for equations with drift terms, conditions on the uniqueness of stationary solutions were obtained. Furthermore, I considered bistable reaction-diffusion equations on the plane and found a solution such that its interface locally approaches a line while the position of the interface gets away from that of a planar travelling wave solution in the direction of travel.

Free Research Field

数物系科学

Academic Significance and Societal Importance of the Research Achievements

反応拡散方程式に対する非定数定常解の存在と安定性の研究は、パターン形成に関する数学的研究として最も関心の高い研究の１つである。しかし、定常解構造を決定することは、特に領域の形状が複雑な場合には非常に困難な問題となる。本研究では新しいタイプの領域の特異極限を考え、詳細な解析が可能な方程式へ問題を帰着させることでこれを克服した。この方法を用いることで、さらに複雑な領域において解構造の解析が可能となると期待される。