Mechanism of singularity preservation for solutions in parabolic equations

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14567
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tokyo Institute of Technology
• Principal Investigator: Takahashi Jin 東京工業大学, 情報理工学院, 助教 (40813001)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 半線形熱方程式 / fast diffusion方程式 / 多孔質媒体方程式 / 特異解 / 臨界指数 / 可解性 / 山辺流

Outline of Final Research Achievements

For the semilinear heat equation (a typical example of a nonlinear parabolic partial differential equation), we constructed solutions with a time-dependent singularity and specified the behavior of the solutions near the singular point. Moreover, in view of the loss of singularity, we also studied initial value problems and initial boundary value problems. Then, we obtained sharp conditions for the solvability of the problems.
For the fast diffusion equation (an example of a parabolic equation with nonlinear diffusion), we constructed new types of singular solutions such as snaking singularity and anisotropic singularity. Furthermore, we found a relation between the blow-down of singularity in the fast diffusion equation and the disappearance of completeness in the corresponding geometric flow.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

偏微分方程式論における解の特異性はこれまで盛んに研究されてきた．しかし，特異性を保持する解や，その位置が時間依存して動くようなものはあまり扱われてこなかった．本研究においては非線形放物型偏微分方程式の典型例に対し，ある種の臨界的状況において特異解を構成し解析するとともに幾何との関連も得ている．それゆえ，特異解の研究を大きく進展させ，さらに広がりを与えたという学術的意義がある．