Research on the global structure of solutions and their stability for nonlocal boundary value problems by using elliptic functions

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03593
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Ryukoku University
• Principal Investigator: Yotsutani Shoji 龍谷大学, 公私立大学の部局等, 研究員 (60128361)
• Co-Investigator(Kenkyū-buntansha): 森田 善久 龍谷大学, 公私立大学の部局等, 研究員 (10192783) ; 川上 竜樹 龍谷大学, 先端理工学部, 教授 (20546147)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 非線形境界値問題 / 完全楕円積分 / 楕円関数 / 交差拡散方程式 / 反応拡散方程式 / 極限方程式 / 非局所 / 線形化固有値問題

Outline of Final Research Achievements

For typical boundary value problems where the nonlocal term is a definite integral of a solution, we combine and apply classical elliptic functions theory and modern nonlinear PDE theories to clarify the structure of all stationary solutions. We have developed unique methods for it and analyzing the stability.
Especially, according to our research plan, we clarified a limiting equation of the KST cross-diffusion equation, a mathematical model of the expression of cell polarity, and the global structure of the steady-state solution of the Allen-Cahn equation including nonlocal terms, and analyzed their stability.
Furthermore, we have obtained the characterization of the global bifurcation diagram of all stationary solutions of the one-dimensional mathematical model that describes the solidification phenomenon of alloys, which had remained unsolved for about 20 years since the model was proposed, and we have now advanced to stability analysis.

Free Research Field

数理解析学関連

Academic Significance and Societal Importance of the Research Achievements

従来,微分方程式の解の存在のための条件を求めたり,局所的な解の分岐構造に対して,非線型偏微分方程式論をはじめとして数学的研究がなされて現在も発展を続けている.ところが，解の精密形状を知ることや解の大域的分岐構造解明はより困難な問題である．
しかし,生命現象や物理現象等にあらわれる数理モデルに対して,数学的な結果を利用できるようにするためには,困難であるが是非克服すべき問題である.
我々は基本的で典型的な微分方程式で記述される数理モデルに対し,特異摂動問題の解の精密な陽的表示式,極限形状の精密な表示式,２次分岐等を含めた解の大域的分岐構造を得るための独自の方法を開発・発展させた．