Compact traveling waves for mean-curvature flow with driving force

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K13458
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Okayama University
• Principal Investigator: MONOBE HARUNORI 岡山大学, 異分野基礎科学研究所, 特任准教授 (20635809)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 平均曲率流方程式 / 進行波解

Outline of Final Research Achievements

We studied the mean-curvature flow with driving force which is derived from some mathematical models, such as cell-locomotion and droplet motion. As a result, we have the following results : (1) if the driving force is positive, there exists a unique compact traveling wave (CTW) solutions. (2) if the driving force is negative, there does not exists any CTW solutions. (3). if the driving force is sign-changing and some conditions are satisfied, there exists a unique CTW solutions. (4). if CTW exists, every CTW is convex and unstable.

Free Research Field

数学

Academic Significance and Societal Importance of the Research Achievements

偏微分方程式で記述される数理モデルにおいて、スポット状パターンの発生メカニズムは未だ明確にはわかっておらず、現在もモデル構築の際は、経験則に頼るところが多い。このため、より複雑な形状のパターンへの解析が滞っている。本研究成果は、スポット状の物質が移動する現象の数理モデルを考えた時に、その構築方法や解の振る舞いの解析の手助けになると考えられる。