2018 Fiscal Year Final Research Report

Study of second order elliptic operators with unbounded coefficients

Research Project

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Project/Area Number	16K17619
Research Category	Grant-in-Aid for Young Scientists (B)
Allocation Type	Multi-year Fund
Research Field	Basic analysis
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)
Research Collaborator	Metafune Giorgio Spina Chiara Wakasugi Yuta Ikeda Masahiro Yoshii Kentarou
Project Period (FY)	2016-04-01 – 2019-03-31
Keywords	2階楕円型作用素 / 非有界な係数 / 半線形熱方程式 / 消散型波動方程式 / テスト関数法
Outline of Final Research Achievements	The generation of analytic semigroups, the structure of heat kernel, and the spectrum of second order elliptic operators with unbounded coefficients (with a special structure) in Lp-spaces are established. Also, weighted energy estimates and diffusion phenomena for wave equations with space-dependent damping term are discussed. Moreover, methods for proving blowup phenomena for semilinear heat equations, Schr"odinger equations and damped wave equations are refined. The blowup phenomena for semilinear wave equations with scale-invariant space(time) dependent damping terms are also found.
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。この研究は、その現象がどういうものであるかにかかわらず、記述された方程式の型のみから得られる普遍的な性質を読み解くことに用いることができる。この研究によって、特異性をもつ現象で今まで扱いきれていなかった現象を解析できる可能性が高まったと考えている。