2022 Fiscal Year Final Research Report
Structural conditions for global existence of solutions and the asymptotic behavior of global solutions for systems of nonlinear partial differential equations related to nonlinear waves
| Project/Area Number |
18H01128
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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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 | Osaka University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | 非線形波動方程式系 / 非線形シュレディンガー方程式系 / 非線形クライン=ゴルドン方程式系 / 大域解 / 漸近挙動 / 零条件 / 弱零条件 |
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| Outline of Final Research Achievements |
We have extended the global existence results for systems of semilinear wave equations with single-speed under some "weak" null conditions to systems of quasi-linear wave equations in two and three space dimensions, and to systems of semilinear wave equations with multiple-speed in two space dimensions. We have also obtaind global existence for systems of semilinear wave equations, and systems of wave and Klein-Gordon equations in three space dimensions under a kind of "weak" null condition with some technical additional assumptions. We examined the asymptotic behavior of global solutions under "weak" null conditions for the above mentioned systems, as well as systems of nonlinear Schroedinger equations.
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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 |
いくつかの偏微分方程式系に対して, 従来よりも弱い条件下での大域解の存在定理を得ることができた. また, 大域解の漸近挙動についても研究し, 小さい初期値の場合であっても, 解が自由解に漸近する以外に, エネルギーが増加もしくは減少したり, 特定の成分にエネルギーが集中したりするなど様々な挙動が起こりうることが明らかになった. また従来は解が自由解に漸近するかどうかに興味がもたれていたが, 自由解に漸近する場合にも, 自由解の初期値が元の初期値とはかけ離れたものになる現象が起こり得ることが明らかになった. これらの結果は非線形偏微分方程式の理解に新たな知見を与えている.
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