Studies on the thoery of elliptic operators with unbounded coefficients and applications

2021 Fiscal Year Final Research Report

• Project/Area Number: 18K13445
• 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 University of Science
• Principal Investigator: Sobajima Motohiro 東京理科大学, 理工学部数学科, 講師 (20760367)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Keywords: 非有界な係数をもつ楕円型作用素 / 消散型波動方程式 / 漸近展開

Outline of Final Research Achievements

In this research, we focus our attention to the development and applications to the theory of second order elliptic operators with unbounded coefficients. In the flamework of partial differential differential equations, we try to find essentials of linear problems and to apply it to the corresponding nonlinear problems.
One of the achievement in the linear problem is the Rellich inequalities in bounded domains. We found the necessary and sufficient conditions on the validity of Rellich inequalities with the Laplacian perturbed by the singular lower order terms.
For the nonlinear problem, we discussed the large time behavior of solutions to linear wave equations with space-dependent damping terms, blowup phenomena and global existence for the corresponding nonlinear problems.

Free Research Field

偏微分方程式論

Academic Significance and Societal Importance of the Research Achievements

非有界な係数をもつ2階楕円型作用素は様々な自然現象を記述する際に用いられる。この研究で、空間変数に依存する消散型波動方程式の長時間挙動に非有界な拡散構造を見ることができた。これは既存の研究からは得られない知見であり、同種の現象が他のモデルにも内在する可能性を示唆している。このことから、今後さらに非有界な係数という枠組みの重要性が高まったと考えている。