Studies on the qualitative theory and singularities of nonlinear partial differential equations

2020 Fiscal Year Final Research Report

• Project/Area Number: 16H02151
• Research Category: Grant-in-Aid for Scientific Research (A)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Meiji University (2017-2020); The University of Tokyo (2016)
• Principal Investigator: Matano Hiroshi 明治大学, 研究・知財戦略機構(中野), 特任教授 (40126165)
• Co-Investigator(Kenkyū-buntansha): 宮本 安人 東京大学, 大学院数理科学研究科, 准教授 (90374743)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 非線形偏微分方程式 / 定性的理論 / 進行波 / 界面運動 / 解の爆発 / 特異極限 / 自由境界問題 / 力学系

Outline of Final Research Achievements

We studied qualitative properties of solutions of nonlinear diffusion equations, with emphasis on the front propagation phenomena. Here are the main achievements.

(1) We succeeded in determining the spreading speed of fronts in predator-prey reaction-diffusion models. (2) We studied disease-spreading models on spatially inhomogeneous media and revealed the directional dependence of the spreading speed. (3) We showed that the shape of the spreading front of an anisotropic diffusion equation approaches the Wulff shape associated with the anisotropy of the diffusion. (4) We classified the dynamics of solutions of semilinear diffusion equations on R. Interesting results have also been obtained for equations with time-periodic or spatially periodic nonlinearities. (5) We considered the singular-limit of volume-preserving Allen-Cahn equations and studies in detail the initial formation of interfaces.

Free Research Field

非線形解析学

Academic Significance and Societal Importance of the Research Achievements

非線形拡散方程式に現れる進行波や波面の広がり現象は，数理生態学や物理学など応用上の観点からも重要であり，近年盛んに研究が行われている．しかし比較定理が成り立たない連立系の場合は，多くが未解明である．本研究では，捕食者被食者系（predator-preyモデル）における波面の広がり現象を研究し，波面の広がりが，確定した速度で起こることを初めて明らかにした．これは，反応拡散系における波面の伝播現象の研究に一つの突破口を開くものである．また，空間１次元半線形拡散方程式のダイナミクスの研究では，交点数非増大則を巧妙に用いた新しい論法を確立した．この手法は，今後，他の研究にも大きく役立つと思われる．