2018 Fiscal Year Final Research Report
The pursuit of new phenomena and methods for supercritical elliptic PDEs
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 |
Section | 一般 |
Research Field |
Mathematical analysis
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Research Institution | The University of Tokyo |
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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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 |
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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Free Research Field |
偏微分方程式
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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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