Nonlinear partial differential equation with a dynamical boundary condition

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K17629
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Ryukoku University (2017-2019); Osaka Prefecture University (2016)
• Principal Investigator: Kawakami Tatsuki 龍谷大学, 理工学部, 教授 (20546147)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 動的境界条件 / 可解性 / 臨界指数 / 拡散極限

Outline of Final Research Achievements

We consider some nonlinear partial differential equations with a dynamical boundary condition in unbounded domains.For the nonlinear elliptic equation in the exterior of unit ball, we gave several results on existence, nonexistence and large time behavior of small positive solutions.For the heat equation, we considered two cases, the half space and the exterior of unit ball, and proved the existence of global-in-time solutions. Furthermore, we showed that, if the diffusion coefficient tends to infinity (it called by the large diffusion limit), then the solution converge to solutions of the Laplace equation with same dynamical boundary condition.Moreover, we also considered some diffusion equations, which are related with anomalous diffusion, and obtained the critical exponent for the existence of global-in-time solutions.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

動的境界条件は近年、国内外において領域の有界性にかかわらず活発に研究が行われてきている。その中で解の時間大域可解性や内部の拡散現象の極限を考察した熱方程式の拡散極限は、時間大域挙動に対する内部と外部の拡散現象の影響の考察や、今後の非線形問題への応用に向けて最も基本的かつ重要な問題であり、必要不可欠な研究と言える。本成果が足掛かりとなり、今後非線形問題等に大いに進展していくことが期待される。