Nonlinear partial differential equation with a dynamical boundary condition
| Project/Area Number |
16K17629
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Mathematical analysis
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| Research Institution | Ryukoku University (2017-2019)
Osaka Prefecture University (2016) |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
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| Keywords | 動的境界条件 / 可解性 / 臨界指数 / 拡散極限 / 外部領域 / 指数型非線形項 / Cartan-Hadamard多様体 / 退化拡散方程式 / 分数冪拡散方程式 / 時間大域可解性 / 高次漸近展開 / 特異拡散方程式 / 非線形楕円型方程式 / 粘性Hamilton-Jacobi方程式 / 指数型非線形 / 半線形楕円型方程式 |
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| 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.
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| 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。
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Report
(5 results)
Research Products
(56 results)
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[Book] ベクトル解析入門2019
Author(s)
壁谷 喜継、川上 竜樹
Total Pages
240
Publisher
共立出版
ISBN
4320113756
Related Report
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