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Study of second order elliptic operators with unbounded coefficients

Research Project

Project/Area Number 16K17619
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Basic analysis
Research InstitutionTokyo 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
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords2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 / 非有界な係数をもつ楕円型作用素 / 波動方程式 / 藤田型方程式 / 非線形消散型波動方程式 / 藤田型熱方程式 / 作用素論 / 関数解析 / 偏微分方程式論 / 解析学基礎 / 関数解析学
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.

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。

Report

(4 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (46 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (3 results) Journal Article (12 results) (of which Int'l Joint Research: 3 results,  Peer Reviewed: 11 results,  Open Access: 2 results,  Acknowledgement Compliant: 2 results) Presentation (31 results) (of which Int'l Joint Research: 4 results,  Invited: 18 results)

  • [Int'l Joint Research] University of Salento(イタリア)

    • Related Report
      2018 Annual Research Report
  • [Int'l Joint Research] University of Salento(イタリア)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Salento(イタリア)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Elliptic and parabolic problems for a class of operators with discontinuous coefficients2019

    • Author(s)
      Giorgio Metafune, Motohiro Sobajima, Chiara Spina
    • Journal Title

      ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA - CLASSE DI SCIENZE

      Volume: 19 Pages: 1-54

    • DOI

      10.2422/2036-2145.201605_002

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain2019

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Journal Title

      Journal of Mathematical Analysis and Applications

      Volume: 470 Issue: 1 Pages: 318-326

    • DOI

      10.1016/j.jmaa.2018.10.004

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method2019

    • Author(s)
      M.Ikeda and M.Sobajima
    • Journal Title

      Nonlinear Analysis, TMA

      Volume: 182 Pages: 57-74

    • DOI

      10.1016/j.na.2018.12.009

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data2019

    • Author(s)
      Motohiro Sobajima, Yuta Wakasugi
    • Journal Title

      Communications in Contemporary Mathematics

      Volume: 印刷中 Issue: 05 Pages: 1850035-1850035

    • DOI

      10.1142/s0219199718500359

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping2019

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Journal Title

      FUNKCIALAJ EKVACIOJ

      Volume: 印刷中

    • NAID

      130008083382

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Lp-analysis of one-dimensional repulsive Hamiltonian with a class of perturbations2018

    • Author(s)
      Motohiro Sobajima, Kentarou Yoshii
    • Journal Title

      AIMS Mathematics

      Volume: 3 Issue: 1 Pages: 21-34

    • DOI

      10.3934/math.2018.1.21

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity2018

    • Author(s)
      Motohiro Sobajima, Yuta Wakasugi
    • Journal Title

      Advances in Differential Equations

      Volume: 23 Pages: 581-614

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data2018

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Journal Title

      Mathematische Annalen

      Volume: 未定 Issue: 3-4 Pages: 1017-1040

    • DOI

      10.1007/s00208-018-1664-1

    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] An elementary proof of asymptotic behavior of solutions of u′′=Vu2017

    • Author(s)
      Giorgio Metafune, Motohiro Sobajima
    • Journal Title

      Proceedings of Equadiff 2017 Conference

      Volume: - Pages: 369-376

    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Journal Article] Kernel estimates for elliptic opertators with second order discontinuous coefficients2017

    • Author(s)
      Giorgio Metafune, Motohiro Sobajima, Chiara Spina
    • Journal Title

      Journal of Evolution Equations

      Volume: 17 Issue: 1 Pages: 485-522

    • DOI

      10.1007/s00028-016-0355-1

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain2017

    • Author(s)
      M. Sobajima and Y. Wakasugi
    • Journal Title

      AIMS Mathematics

      Volume: 2 Issue: 1 Pages: 1-15

    • DOI

      10.3934/math.2017.1.1

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain2016

    • Author(s)
      M. Sobajima and Y. Wakasugi
    • Journal Title

      J. Differential Equations

      Volume: 261 Issue: 10 Pages: 5690-5718

    • DOI

      10.1016/j.jde.2016.08.006

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] On lifespan estimate for solutions of a semilinear damped wave equation via a test function method2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      Critical exponent and nonlinear evolution equations 2018
    • Related Report
      2018 Annual Research Report 2017 Research-status Report
    • Invited
  • [Presentation] Life-span of blowup solutions to semilinear wave equation with space dependent critical damping2018

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Organizer
      日本数学会2018年度年会函数方程式論分科会
    • Related Report
      2018 Annual Research Report
  • [Presentation] スケール臨界な時間変数に依存した係数を摩擦項に持つ波動方程式の解の最大存在時刻について2018

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Organizer
      日本数学会2018年度年会函数方程式論分科会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 2次元外部領域における半線形熱方程式の解の爆発について2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      第167回愛媛大学解析セミナー
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] On lifespan for two dimensional semilinear heat equation in exterior domain2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      International Workshop on “Fundamental Problems in Mathematical and Theoretical Physics”
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Analytic semigroups generated by scale invariant elliptic operators and related inequalities I2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      RIMS共同研究(グループ型)「スペクトル解析におけるスケール不変構造と摂動論の新展開」
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Analytic semigroups generated by scale invariant elliptic operators and related inequalities II2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      RIMS共同研究(グループ型)「スペクトル解析におけるスケール不変構造と摂動論の新展開」
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] スケール臨界の摩擦項をもつ非線形波動方程式の解の爆発について2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      第26回さいたま数理解析セミナー
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] On a test function method for blowup of solutions to semilinear damped wave equations2018

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Sharp lifespan estimates for solutions to two-dimensional semilinear heat equation in exterior domains2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      日本数学会秋季総合分科会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Test function method for blowup of solutions to semilinear evolution equations in sectorial domain2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      RIMS共同研究(公開型)「非線形発展方程式を基盤とする現象解析に向けた数学理論の展開」
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Blow up solutions of semilinear heat equation2018

    • Author(s)
      Motohiro Sobajima
    • Organizer
      Seminario (University of Salento)
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Life-span of blowup solutions to semilinear wave equation with space dependent critical damping2018

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Organizer
      日本数学会年会函数方程式論分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] スケール臨界な時間変数に依存した係数を摩擦項に持つ波動方程式の解の最大存在時刻について2018

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Organizer
      日本数学会年会函数方程式論分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Diffusion phenomena for the wave equation with space-dependent unbounded damping term2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      5th Italian-Japanese Workshop "Geometric Properties for Parabolic and Elliptic PDE’s
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 空間遠方で増大する摩擦項を伴う波動方程式のエネルギー評価について2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      第81回埼玉大学解析ゼミ
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Weighted energy estimates for wave equation with space-dependent damping term growing at infinity2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      Equadiff 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] 空間に依存する摩擦を伴う波動方程式に対するエネルギー評価について2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      第56回実函数論・函数解析学合同シンポジウム
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Weighted energy estimates for damped wave equation with slowly decaying initial data2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      第8回 拡散と移流の数理
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity2017

    • Author(s)
      Motohiro Sobajima, Yuta Wakasugi
    • Organizer
      日本数学会秋季総合分科会函数方程式論分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data2017

    • Author(s)
      Motohiro Sobajima, Yuta Wakasugi
    • Organizer
      日本数学会秋季総合分科会函数方程式論分科会
    • Related Report
      2017 Research-status Report
  • [Presentation] On L^p-theory for second order elliptic operators with unbounded diffusion coefficients2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      Global properties in potential theory of continuous and discrete spaces
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Diffusion phenomena for wave equation with space-dependent damping term2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      Seminario (University of Salento)
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 空間変数に依存するスケール臨界な摩擦項をもつ非線形波動方程式の解の爆発について2017

    • Author(s)
      Motohiro Sobajima
    • Organizer
      大阪大学 微分方程式セミナー
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] Blowup phenomena for semilinear wave equation with space-dependent singular damping2017

    • Author(s)
      Masahiro Ikeda, Motohiro Sobajima
    • Organizer
      第43回発展方程式研究会
    • Related Report
      2017 Research-status Report
  • [Presentation] |x|^αΔを主要項にもつ楕円型作用素が生成する解析半群の積分核評価2017

    • Author(s)
      Motohiro Sobajima, Giorgio Metafune, Chiara Spina
    • Organizer
      日本数学会2017年度年会 関数方程式論分科会
    • Place of Presentation
      首都大学東京 南大沢キャンパス
    • Related Report
      2016 Research-status Report
  • [Presentation] Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain2017

    • Author(s)
      Motohiro Sobajima, Yuta Wakasugi
    • Organizer
      日本数学会2017年度年会 関数方程式論分科会
    • Place of Presentation
      首都大学東京 南大沢キャンパス
    • Related Report
      2016 Research-status Report
  • [Presentation] Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain2016

    • Author(s)
      Motohiro Sobajima
    • Organizer
      The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      フロリダ(アメリカ)
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Kernel estimates for semigroups generated by |x|^αΔ with lower order terms2016

    • Author(s)
      Motohiro Sobajima
    • Organizer
      応用解析研究会
    • Place of Presentation
      早稲田大学 西早稲田キャンパス
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] |x|^αΔを主要項にもつ楕円型作用素が生成する解析半群の積分核評価について2016

    • Author(s)
      Motohiro Sobajima
    • Organizer
      神楽坂解析セミナー
    • Place of Presentation
      東京理科大学 神楽坂キャンパス
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] An embedding estimate for the repulsive Hamiltonian in L^p2016

    • Author(s)
      Kentarou Yoshii, Motohiro Sobajima
    • Organizer
      日本数学会2016年度秋季総合分科会 実函数論分科会
    • Place of Presentation
      関西大学 千里山キャンパス
    • Related Report
      2016 Research-status Report

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Published: 2016-04-21   Modified: 2020-03-30  

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