Spatio and temporal dynamics of solutions for reaction-diffusion equations with nonlocal effect 非局所効果を持つ反応拡散方程式における解の時空間ダイナミクスについて

収集根拠 : 博士論文（自動収集）

資料形態 : テキストデータ

コレクション : 国立国会図書館デジタルコレクション > デジタル化資料 > 博士論文

The pattern formation problem is one of the most fascinating and essential problems in the natural sciences. In recent years, the study of pattern formation has been theoretically analyzed using reaction-diffusion equations with nonlocal effect described by convolution with the appropriate integral kernel as mathematical models in various fields such as biology, material science, and medicine. Therefore, the mathematical analysis of the behavior of solutions in reaction-diffusion equations with nonlocal effect is becoming more and more important with each passing year. This thesis focuses on developing new analytical methods for theoretically considering the spatio and temporal dynamics of solutions and their applications to understand how nonlocal effect affects the pattern formation process.First, this thesis introduces the reaction-diffusion equation and mathematical models with nonlocal effect in Section 1. Then, it explains the effectiveness of mathematical modeling with nonlocal effect and its relationship with the reaction-diffusion equation. After that, as the first result, a new method to show the existence of traveling wave solutions is described, and its applications are presented in Section 2. In addition, as the second result, a method for analyzing weak interactions between localized patterns, such as stationary and traveling wave solutions, is explained, and its applications are given in Section 3. Finally, as the third result, we consider the asymptotic behavior of the zero points of solutions to the diffusion equation and the fractional diffusion equation in Section 4, with the motivation to analyze the nonlocal effect on the spatial propagation mechanism of substances.

(主査) 教授 栄 伸一郎, 教授 神保 秀一, 准教授 黒田 紘敏, 准教授 田﨑 創平

理学院（数学専攻）

The pattern formation problem is one of the most fascinating and essential problems in the natural sciences. In recent years, the study of pattern formation has been theoretically analyzed using reaction-diffusion equations with nonlocal effect described by convolution with the appropriate integral kernel as mathematical models in various fields such as biology, material science, and medicine. Therefore, the mathematical analysis of the behavior of solutions in reaction-diffusion equations with nonlocal effect is becoming more and more important with each passing year. This thesis focuses on developing new analytical methods for theoretically considering the spatio and temporal dynamics of solutions and their applications to understand how nonlocal effect affects the pattern formation process. First, this thesis introduces the reaction-diffusion equation and mathematical models with nonlocal effect in Section 1. Then, it explains the effectiveness of mathematical modeling with nonlocal effect and its relationship with the reaction-diffusion equation. After that, as the first result, a new method to show the existence of traveling wave solutions is described, and its applications are presented in Section 2. In addition, as the second result, a method for analyzing weak interactions between localized patterns, such as stationary and traveling wave solutions, is explained, and its applications are given in Section 3. Finally, as the third result, we consider the asymptotic behavior of the zero points of solutions to the diffusion equation and the fractional diffusion equation in Section 4, with the motivation to analyze the nonlocal effect on the spatial propagation mechanism of substances.