Asymptotic behavior of solutions to hyperbolic and dispersive equations with damping terms

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03596
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Fukuoka Institute of Technology
• Principal Investigator: Takeda Hiroshi 福岡工業大学, 工学部, 教授 (10589237)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 消散型波動方程式 / 漸近形 / 時間減衰評価 / 平滑化

Outline of Final Research Achievements

For initial value problems of hyperbolic and dispersive equations with a damping term in the linear principal part, the asymptotic behaviors of the solutions, especially the time-decay estimates and the identification of the asymptotic profile, are obtained. The high frequency parts of the linear solution are estimated according to the difference in the damping terms, and a smoothing effect is derived. Based on the fact, together with the estimates which suggest the dissipation of the linear solutions, asymptotic behavior of time-global solutions to small initial values of nonlinear problems are described sharply.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

消散項を含む双曲型・分散型方程式の初期値問題の対応する放物型方程式や、消散項を外した方程式には解の漸近挙動に対する精密な理論がよく知られている。本研究課題はその応用として得られる帰結ではなく、線形主要部のすべての項を用いた方程式固有の性質を見直して消散項を含む双曲型・分散型方程式に特化した理論体系を構築することを目指している。