The analysis of effects by derivative loss in the fundamental solution of the high-dimensional wave equation on nonlinear problems

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K04964
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Tohoku University (2018); Future University-Hakodate (2015-2017)
• Principal Investigator: Takamura Hiroyuki 東北大学, 理学研究科, 教授 (40241781)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 波動方程式 / 非線形 / 初期値問題 / 微分損失 / 時間減衰 / 時間大域存在 / 有限時間爆発 / 高次元

Outline of Final Research Achievements

In the high dimensional space-time, which means larger dimension than 3 in space, the fundamental solution of the wave equation has the so-called "derivative loss". It is one of the reason of preventing us to analyze the nonlinear problem by means of point-wise estimates of the solution. In this research, we clarify that the derivative loss in the nonlinear term of the integral equation which is equivalent to the differential equation with the small data and the power-type nonlinearity does not make any influence to the maximal existence time of the solution. In addition, we have an extra non-local term of time-derivative of the nonlinear term in the differential equation if such a derivative loss is neglected. Moreover, we also clarify that the derivative loss in the linear term contributes the main time-decay, and makes the critical exponent dividing global-in-time existence and finite-time blow-up go the solution smaller if it is neglected.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

空間次元が３までの時空間における様々な現象は、実体験として容易に想像し得るものであるが、数学や物理ではそれ以上の高次元空間を考えて事象を解析することが自然になりつつある。特に、自然現象を記述する偏微分方程式を数学で統一的かつ厳密に扱う場合、空間次元を一般化しておくことは重要な課題の一つとなっている。本研究はその中で、ほとんど解明されてこなかった高次元波動方程式の解の性質を、あえて困難な各点評価を用いることによって明らかにした。この成果は今後、より実際の現象に近い摩擦現象を考慮した消散項付き波動方程式に応用されたり、その分野の研究の位置付けの中で重要な存在になると思われる。