2023 Fiscal Year Final Research Report
Study on blowup phenomena for Shcr\"odinger equations with non-gauge invariant power type nonlinearities
| Project/Area Number |
20K14337
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Ryukoku University (2023)
Nagoya University (2020-2022) |
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Principal Investigator |
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| Project Period (FY) |
2020-04-01 – 2024-03-31
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| Keywords | 爆発解析 / 初期値問題 / 分散型方程式 |
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| Outline of Final Research Achievements |
In this study, we have completed the classification of integrable initial conditions for the global time solvability of solutions to the one-dimensional semi-linear Schr\"odinger equation under periodic boundary conditions. Particularly, a singular instability concerning initial conditions has been identified, wherein finite-time blow-up phenomena of solutions occur due to oscillations in the initial state, no matter how small they are. Additionally, in the case of two dimensions, a significant inhibitory effect on solutions due to interactions between frequency components has been observed, suggesting the potential stabilization of initial value dependency of solutions in higher dimensions.
Furthermore, we have relaxed the technical requirements for evaluating the maximum existence time of solutions to semi-linear damped wave equations and derived a precise evaluation of the maximum existence time of solutions when the integral average of the initial state is zero.
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| Free Research Field |
偏微分方程式論
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| Academic Significance and Societal Importance of the Research Achievements |
本研究における学術的な意義は、従来の解析手法では不可欠であった初期状態に対する積分平均の符号条件を緩和した事にある。従来の解析手法は、複雑な非線型効果による解の増大の様相を、解の積分量に着目する事で要約していた。一方で、本研究ではシュレーディンガー方程式や消散型波動方程式の初期状態の振動の影響を考慮する事で、方程式の非線型効果に対して一歩踏み込んだ理解を与えた。特に、本研究で解明した初期状態の振動が与える解の挙動への影響は、計算シュミレーションによる観測が困難なものであり、純粋数学による解析手法の効果を示す一例を提示できた。
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