Geometric structure of manifold and the blow-up problem of nonlinear heat equation
| Project/Area Number |
23740128
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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 |
Global analysis
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| Research Institution | Okayama University of Science (2013-2014)
Hokkaido University (2012)
Meiji University (2011) |
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Principal Investigator |
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| Project Period (FY) |
2011-04-28 – 2015-03-31
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| Project Status |
Completed (Fiscal Year 2014)
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| Budget Amount *help |
¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2014: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2013: ¥520,000 (Direct Cost: ¥400,000、Indirect Cost: ¥120,000)
Fiscal Year 2012: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
Fiscal Year 2011: ¥390,000 (Direct Cost: ¥300,000、Indirect Cost: ¥90,000)
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| Keywords | 爆発問題 / 非線形熱方程式 / 複素藤田型方程式 / 爆発点の制御 / 曲率流 / 自由境界問題 / 特異点 / 爆発 / 曲線短縮流 / 自由境界値問題 / 移流項付き曲率流 / 漸近自己相似性と収束 / 特異点解析 / 漸近挙動と分類定理 / 体積保存 / 非線形放物型方程式 / 平均曲率流 / 反応拡散方程式 / CLM方程式 / 非線形熱方程式の爆発問題と多様体の幾何構造 / 多様体上の極小曲面と変分法 / 平均曲率流の自由境界値問題 |
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| Outline of Final Research Achievements |
The aim of this research project is to consider nonlinear problems related to interfacial motions and blow-up phenomena.The methods are based on scaling related to the symmetry of the equation and on the theory of infinite dimensional dynamical systems. The following results have been obtained:(1)We studied nonlinear heat equations with power nonlinearity multiplied by a spatial inhomogeneous coefficient, which has zero points in the domain.We have given several conditions to determine blow-up point point.(2)We consider the Cauchy problem for a system of parabolic equations which is derived from a complex-valued equation with a quadratic nonlinearity. Global dynamics and blow-up phenomena was considered.(3)We have succeeded in determining all possible type of the behavior of a free boundary problem of a curvature-dependent motion of a curve, by applying new ``extended'' intersetion number principle. The results for asymptotic behavior and convexity are also established.
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Report
(5 results)
Research Products
(17 results)
