Shapes of functions, asymptotic behavior and function spaces associated with solutions to nonlinear dispersive equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K05311
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Chiba University
• Principal Investigator: Sasaki Hironobu 千葉大学, 大学院理学研究院, 准教授 (00568496)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 非線型分散型方程式 / 散乱作用素 / ソボレフ空間

Outline of Final Research Achievements

In this study, we consider scattering problems for nonlinear dispersive equations. We obtain the following results:
(1) We considered the 3-dim. Klein-Gordon equations whose nonlinearity is cubic, and we proved that the scattering operator maintain the smoothness and decay of the input data. (2) We considered the 2-dim. Klein-Gordon equations whose nonlinearity behaves like u(exp(|u|^2)-1), and we proved that the scattering operator maintain the smoothness and decay of the input data. (3) We considered the semi-relativistic equation whose interaction potential satisfies some suitable conditions, and we proved that the scattering operator maintain the smoothness and decay of the input data.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

本研究で得られた成果を分かりやすく述べると、「入力データとして与えた関数（実験の世界では粒子に相当）が滑らかだったり、遠方で減衰するものであったら、滑らかな非線型相互作用によって変化した出力データも同程度以上の滑らかさや減衰性をもつことを示した」となる。純粋数学的には散乱の逆問題に応用が可能と思われる。また、粒子の実験を行う際の有用なヒントにもなりうる。