Singular limit problem for nonlinear PDE and interface motion coupled with potentials

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05275
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Okayama University
• Principal Investigator: Oshita Yoshihito 岡山大学, 自然科学研究科, 准教授 (70421998)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 非線形偏微分方程式

Outline of Final Research Achievements

We show linear stability or instability for radially symmetric equilibrium solutions to the system of interface equation and two parabolic equations arising in the singular limit of three-component activator-inhibitor models.

Also, we study the free boundary problem describing the micro phase separation in the regime that one component has small volume fraction ρ such that the micro phase separation results in an ensemble of small disks of one component. We rigorously derive the heterogeneous mean-field equations on a time scale of the order of R^3 ln (1/ρ), where R is the mean radius of disks. On this time scale, the evolution is dominated by coarsening and stabilization of the radii of the disks, whereas migration of disks becomes only relevant on a larger time scale.

Free Research Field

非線形偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

変分原理，リャプノフ・シュミットの縮約法，線形化安定性解析，不変多様体への縮約理論，均質化の手法，漸近展開法などの手法を用いたり，新しい数学的手法の開発をすることで，反応拡散系の特異極限問題，界面方程式の解の構造，非線形楕円型偏微分方程式の特異極限問題に現れる様々な解の集中現象の数理解析，自然界に現れるパターン形成の数理的構造や非線形偏微分方程式の数理解析の新しい知見を得ることができる。