Characterization of hyperbolic operators with the coefficients of the principal part depending only on the time variable for which the Cauchy problem is well-posed

2020 Fiscal Year Final Research Report

• Project/Area Number: 16K05222
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: University of Tsukuba
• Principal Investigator: Wakabayashi Seiichiro 筑波大学, 数理物質系(名誉教授), 名誉教授 (10015894)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: 双曲型作用素 / コーシー問題 / Ｃ∞適切性 / 超局所解析

Outline of Final Research Achievements

In the preceding researches I obtained sufficient conditions of C∞ well-posedness of the Cauchy problem for higher-order hyperbolic operators with double characteristics satisfying the conditions that the coefficients of the principal parts are real analytic functions of the time variable. And I showed that these sufficient conditions are also necessary when the space dimension is less than 3 or the coefficients of the principal parts are semi-algebraic functions ( e.g., polynomials ) of the time variable.
I also considered the Cauchy problem for higher-order hyperbolic operators with triple characteristics whose coefficients are real analytic functions of the time variable. And I obtained similar results concerning the characterization of C∞ well-posedness.

Free Research Field

数学・基礎解析学

Academic Significance and Societal Importance of the Research Achievements

双曲型作用素に対するコーシー問題の C∞適切性の特徴付けは、偏微分方程式論における主要なテーマの１つであり、これまでに多くの研究があるが、未だ満足のいく結果は得られていないのが現状である。報告者が、主部の係数が時間変数にのみに依存する特別な枠組みではあるが、２重特性的である場合に C∞適切性の特徴付けを与えたことは、今後のこの分野の研究・発展に貢献するものと期待される。また３重特性的な場合を扱うために、subprincipal symbol を一般化して、sub-sub-principal symbol を初めて定義して、係数が時間変数のみに依存する場合に、C∞適切性の特徴付けを与えた。