Global solutions to the Cauchy problem for systems of quasi-linear wave equations satisfying the weak null condition

2023 Fiscal Year Final Research Report

• Project/Area Number: 21K03324
• 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: Mie University
• Principal Investigator: Hidano Kunio 三重大学, 教育学部, 教授 (00285090)
• Co-Investigator(Kenkyū-buntansha): 横山 和義 北海道科学大学, 工学部, 教授 (20316243)
• Project Period (FY): 2021-04-01 – 2024-03-31
• Keywords: 非線形波動方程式 / 初期値問題 / 時間大域解 / 零条件

Outline of Final Research Achievements

The Cauchy problem for a 2-speed and 3-component semi-linear system of wave equations has been studied in three space dimensions. The standard null condition, which is a sufficient condition for global existence of solutions with small data, is violated for the system, and hence a loss of time decay occurs in a certain component. We may reasonably expect that some gain of time decay will occur in the nonlinear interaction between such component and the other one with different propagation speed. In fact, it is technically quite difficult to observe such gain of time decay. At the cost of assuming radial symmetry of the equations and the data, we have succeeded in showing global existence of small, radial solutions for small, radial data. A certain related problem in one space dimension has been also studied.

Free Research Field

非線形偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

空間3次元における非線形波動方程式系の初期値問題が, 小さくなめらかな初期値に対して時間大域解をもつための非線形項の形状に関する条件として知られる零条件は, あくまでも十分条件であり必要条件ではない. そこで, 零条件が破綻しているものの, それでも小さくなめらかな初期値に対して時間大域解が存在するような非線形項にはどのようなものがあるのかを追求する方向で研究を進めてみた.