Challenges to unexplored fields of research on the Cauchy problem for systems of quasi-linear wave equations--large-time behavior and regularity of solutions--

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K03365
• 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)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: wave equation / null condition / weak null condition / global existence / blow up / combined effect

Outline of Final Research Achievements

The Cauchy problem for quasi-linear systems of wave equations has been studied
under the condition that the null condition or the weak null condition is satisfied. Providing that the null condition is satisfied, we have relaxed the decay condition on the data which Christodoulou assumed for global existence of small solutions. Besides, concerning the problem under the weak null condition, we have found that some cubic terms, ``mixed'' with a certain quadratic term, are in fact ``critical''. That is, some cubic terms become a serious hurdle for the proof of global existence result, although they are higher-order terms.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

2次および3次以上の高次の項の非線形項にもつ, 空間3次元における非線形波動方程式の時間空間大域的な解の存在・非存在を考察することは, 幾何学や数理物理学に現れる非線形方程式の解の存在・非存在への応用があり, 偏微分方程式論において重要な問題になる. 2次の項がnull conditionまたはweak null conditionを満たす場合が, 大域的な解の存在を目標とする立場からは大変に重要で, 今回の研究期間中に得られた諸結果は国際誌に発表されている. 既に海外の研究者による別のアプローチが考案されており, この方面の研究で存在感を放つ基本的な文献になるものと期待される.