2019 Fiscal Year Final Research Report
A study of asymptotic behavior of threshold orbit arising in gradient systems with noncompact energy structure
| Project/Area Number |
17K05323
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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 | Osaka University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Keywords | 臨界型偏微分方程式 / プロファイル分解 / コンパクト性の破れ / 時間大域解 / 変分問題 |
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| Outline of Final Research Achievements |
We treat the time-global asymptotic behavior of some partial differential equations with non-compact solution orbits in phase space. We also deal with the solubility of the maximization problem in order to analyze the non-compactness situation for the functional inequalities associated with these partial differential equations.
For semi-linear parabolic equations, it is shown that the norm of the solution is bounded along the time-global solution, and that the solution has an asymptotic behavior over time, represented by a superposition of "bubbles" in which a finite number of equilibrium solutions in the whole space are deformed under the non-compact group action associated with each of them; for maximization problems associated with Trudinger-Moser-type inequalities, it is shown that the solubility of the maximization problem varies sensitively depending on the size of the lower-order term, which acts to break the compactness of the function, and the size of the constraint parameters.
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| Free Research Field |
非線型解析
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| Academic Significance and Societal Importance of the Research Achievements |
従来の力学系理論は、解軌道が相空間の中で相対コンパクトな場合、解のオメガ極限集合が力学系の平衡点の集合の部分集合になることを主張するものであったが、本来偏微分方程式論など具体的な理論での興味は「有界な解軌道」の漸近挙動である。従って、従来の力学系理論における軌道のコンパクト性の仮定は、強すぎる仮定とみなすことができる。本研究によって扱われたいくつかの非コンパクト性を内包する偏微分方程式についての結果から、「有界性＋プロファイル分解」パラダイムに基づく解析が有効であることが示された。これにより、既存の抽象力学系の理論を、解軌道が非コンパクトだが有界な場合に拡張する足掛かりを得ることができた。
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