On blow-up solutions for system of nonlinear drift-diffusion equations with nonlocal interactions

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05219
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Muroran Institute of Technology
• Principal Investigator: Kurokiba Masaki 室蘭工業大学, 大学院工学研究科, 教授 (60291837)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 退化型移流拡散方程式系 / 空間高次元 / 多成分移流拡散方程式系 / 有限時間爆発解 / スケール臨界関数空間 / 特異極限問題 / Lebesgue-Bochner 空間 / 熱方程式の最大正則性

Outline of Final Research Achievements

In this research, we deal with the initial value problem of the degenerate drift-diffusion system with the fast nonlinear diffusion. We have introduced a new weighted Lp space and applied Shannon's inequality to show the new blow-up condition of the solution. Shanonn's inequality depends on the heat capacity ratio, and is an extended version of the conventional inequality. In Second subject, we deal with the singular limit problem of the initial value problem of the Keller-Segel equation in the scale critical function space. It is shown that the strong solution in the scaling critical function space converges to the solution of the parabolic-elliptic drift-diffusion system when the relaxation time is infinite. To prove the singular limit problem, the generalized maximum regularity inequality for the heat equation is applied.

Free Research Field

非線型偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

移流拡散方程式は，半導体，プラズマ粒子の移動現象，中性子星の誕生，生物モデルなどそのサイズスケールが異なりながらも粒子の拡散と凝集の機構で様々な現象共通の数理構造が記述する．移流拡散方程式系の数理構造を明らかにしていくことは普遍的な科学的真理を求めていくことである．また腫瘍モデルの移流拡散方程式系は多成分系でその解析は膨大な情報量を必要とするが，癌の研究に医学的に貢献するものになる．半導体の設計にも移流拡散方程式の研究は大変重要である．