2023 Fiscal Year Final Research Report
Study for nonlinear partial differential equation with Sobolev critical/supercritical nonlinearity
| Project/Area Number |
20K03706
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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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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Tsuda University |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 定在波 / 基底状態 / 閾値解 / 分類 / モース指数 / 非退化性 |
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| Outline of Final Research Achievements |
First result is concerned with the existence of a standing wave for nonlinear Schrodinger equation on the cylinder. We needed technical conditions on the nonlinearity for the existence of the standing waves so far. We succeed to remove the conditions and proved that the standing waves exist.
Second result is about the ground state for double-power nonlinear Schrodinger equations. From the previous studies, we know that there exists a critical frequency for the existence/non-existence of the ground state. However, it was not known whether the ground state exists or not on the critical one. By using the blowup analysis, we found that there exists the ground state in this case.
Finally, we studied the global dynamics of the solutions to the double-power nonlinear Schrodinger equations. We classify the dynamics of solutions which have the same energy of the ground state by the initial data.
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| Free Research Field |
非線形偏微分方程式
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| Academic Significance and Societal Importance of the Research Achievements |
シリンダー上の非線形シュレディンガー方程式の定在波の存在で用いた手法は、他の方程式にも応用が期待できるため、汎用性があるものと思われる。二重べき非線形シュレディンガー方程式の基底状態の存在・非存在については、これまでみられなかった現象が起きることを証明でき、学術的に興味深い。最近では、対応する楕円型方程式の正値解を分類することも出来、今後も進展が期待される。二重べき非線形シュレディンガー方程式の解の大域挙動に関しては、既存の手法とは異なり、one-pass theoremと呼ばれる定理を用いて証明した。この定理が成立すれば、一般の非線形項に対して同様の結果を得られることが期待出来る。
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