| Project/Area Number |
16K05225
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | The University of Tokyo |
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Principal Investigator |
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| Research Collaborator |
Naito Yuki
Wakasa Tohru
Kosaka Atsushi
Takahashi Kazune
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | ソボレフ優臨界 / 半線形楕円型偏微分方程式 / 分岐図式 / 正値特異球対称解 / 一般化相似変換 / 一般の増大度 / Joseph-Lundgren指数 / 交点数 / 楕円型偏微分方程式 / 分岐理論 / 準線形楕円型方程式 / 一般化スケール変換 / 半線形楕円型方程式 / 反応拡散方程式 |
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| Outline of Final Research Achievements |
The structure of the positive solutions of semilinear elliptic equations with supercritical growth in a ball is studied. The solution structure is deeply related to a radial singular solution, and hence it is also studied. When the principal part of the nonlinear term is a pure power function of an exponential function, we show that the classical solution converges to the singular solution in some sense.
On the other hand, if the nonlinear term does not have a algebraic growth or an exponential growth, then we found that the generalized similarity transformation, which was fond by Professor Fujishima of Shizuoka University, is key to study. Using this transformation, we clarify the solution structure in that case.
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| Academic Significance and Societal Importance of the Research Achievements |
半線形楕円型偏微分方程式の研究において，非線形項の増大度がソボレフの意味の，劣臨界もしくは臨界の場合は，膨大な量の研究がなされている．一方，優臨界の場合は，有効な関数解析的手法が存在しないため，その解構造や解の性質等は未知の部分が大きかった．
本研究では，球対称解に制限することによって，常微分方程式の手法を用いて解構造や解の様々な性質を明らかにすることが目的である．本研究によって優臨界方程式に特有の現象が明らかにされ，さらに，対応する放物型方程式の研究に有効な様々な手法や情報を提供した．
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