Study on asymptotic behavior and singularities of solutions for nonlinear dispersive equations by using geometric symmetry

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14342
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Osaka University
• Principal Investigator: Mamoru Okamoto 大阪大学, 大学院理学研究科, 准教授 (40735148)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 適切性

Outline of Final Research Achievements

We study on the singularity and asymptotic behavior of solutions for the nonlinear dispersive equations. We obtain an almost sharp well-posedness result of the Cauchy problem for a system of nonlinear Schrodinger equations. We also study the asymptotic behavior of solutions for a fourth order nonlinear Schrodinger equation with critical nonlinearity in the sense of scattering. Moreover, we prove the ill-posedness of the Cauchy problem for the nonlinear wave equation.

Free Research Field

関数方程式論

Academic Significance and Societal Importance of the Research Achievements

非線形項に微分を含む非線形シュレディンガー方程式において、逐次近似法を用いる限りほとんど最良な適切性を得ることができた。そこで培った手法により、非線形相互作用の制御手法の方針が得られた。また、漸近挙動を解明し、散乱の意味で臨界な状況における解の振る舞いを明らかにした。さらに、非適切性の証明により、解の特異性がいかに発生するかを詳細に調べることができた。