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Analysis on the critical nonlinear wave equations in high dimensions and its applications

Research Project

Project/Area Number 24540183
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionFuture University-Hakodate

Principal Investigator

TAKAMURA HIROYUKI  公立はこだて未来大学, システム情報科学部, 教授 (40241781)

Project Period (FY) 2012-04-01 – 2015-03-31
Project Status Completed (Fiscal Year 2014)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2012: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords非線形波動方程式 / 初期値問題 / 古典解 / 最大存在時間 / 時間大域存在 / 有限時間爆発 / ライフスパン / 高次元空間 / 球対称解 / 解の表示 / 常微分不等式 / 微分損失 / 時間減衰 / 国際研究者交流、イタリア / 国際研究者交流、中国
Outline of Final Research Achievements

The general theory for nonlinear wave equations is devoted to the estimate of the maximal existence time of classical solutions with small data. In high dimensions in which the spatial dimension is grater than 3, its optimality can be discussed only for the quadratic terms in four space dimensions. In such a case, the square of unknown functions itself is known to be a unique example of the nonlinear term which leads to the blow-up of the solution in finite time. The purpose of this study is to look for a criterion which classifies the nonlinear terms including the square of unknown functions into the global existence case or the blow-up case.

Report

(4 results)
  • 2014 Annual Research Report   Final Research Report ( PDF )
  • 2013 Research-status Report
  • 2012 Research-status Report
  • Research Products

    (36 results)

All 2015 2014 2012 Other

All Journal Article (4 results) (of which Peer Reviewed: 3 results,  Open Access: 1 results,  Acknowledgement Compliant: 1 results) Presentation (32 results) (of which Invited: 26 results)

  • [Journal Article] Almost global solutions of semilinear wave equations with the critical exponent in high dimensions2014

    • Author(s)
      H.Takamura & K.Wakasa
    • Journal Title

      Nonlinear Analysis, TMA

      Volume: 109 Pages: 187-229

    • DOI

      10.1016/j.na.2014.06.007

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] 半線形波動方程式入門と最近の話題2014

    • Author(s)
      高村博之
    • Journal Title

      第36回発展方程式若手セミナー報告集

      Volume: 36 Pages: 1-20

    • Related Report
      2014 Annual Research Report
  • [Journal Article] The final problem on the optimality of the general theory for nonlinear wave equations2012

    • Author(s)
      H.Takamura and K.Wakasa
    • Journal Title

      Progress in Mathematics (M.Ruzhansky and M.Sugimoto and J.Wirth eds., "Evolution Equations of Hyperbolic and Schr\"odinger Type")

      Volume: 301 Pages: 315-324

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] The blow-up and lifespan of solutions to systems of semilinear wave equation with critical exponents in high dimensions2012

    • Author(s)
      Y.Kurokawa and H.Takamura and K.Wakasa,
    • Journal Title

      Differential and Integral Equations

      Volume: 25 Pages: 363-382

    • Related Report
      2012 Research-status Report
    • Peer Reviewed
  • [Presentation] 単独非線形波動方程式の一般論とその最適性最終問題2015

    • Author(s)
      高村博之
    • Organizer
      北海道地区数学教育協議会高校サークル3月例会
    • Place of Presentation
      北海道函館工業高等学校(北海道函館市)
    • Year and Date
      2015-03-14
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 加藤の補題の精密化と消散波動方程式への応用2015

    • Author(s)
      高村博之
    • Organizer
      非線形波動方程式論研究会
    • Place of Presentation
      公立はこだて未来大学(北海道函館市)
    • Year and Date
      2015-01-25
    • Related Report
      2014 Annual Research Report
  • [Presentation] 加藤の補題と劣臨界半線形波動方程式への応用2015

    • Author(s)
      高村博之
    • Organizer
      数理物理に現れる偏微分方程式研究会
    • Place of Presentation
      民宿「三五郎」(静岡県賀茂郡)
    • Year and Date
      2015-01-10
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Classical solutions of 4D wave equations with quadratic nonlinearities2014

    • Author(s)
      H. Takamura
    • Organizer
      International Conference on Recent Advances in Hyperbolic Partial Differential Equations
    • Place of Presentation
      広島国際会議場(広島県広島市)
    • Year and Date
      2014-12-04
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Analysis on ordinary differential inequalities arising from nonlinear wave equations2014

    • Author(s)
      H. Takamura
    • Organizer
      RIMS研究集会「実領域における常微分方程式の安定性理論とその応用」
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2014-11-06
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Global existence for semilinear wave equations with the critical blow-up term in high dimensions2014

    • Author(s)
      H. Takamura
    • Organizer
      International Conference on Asymptotic Analysis for Nonlinear Dispersive and Wave Equations
    • Place of Presentation
      大阪大学(大阪府豊中市)
    • Year and Date
      2014-09-09
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The final problem on the optimality of the general theory for nonlinear wave equations and recent topics2014

    • Author(s)
      H. Takamura
    • Organizer
      Sixth Euro-Japanese Workshop on Blow-up
    • Place of Presentation
      東京工業大学(東京都目黒区)
    • Year and Date
      2014-09-02
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 半線形波動方程式入門と最近の話題(特別講演)2014

    • Author(s)
      高村博之
    • Organizer
      第36回発展方程式若手セミナー
    • Place of Presentation
      熊本県休暇村南阿蘇(熊本県阿蘇郡高森町)
    • Year and Date
      2014-08-29 – 2014-08-30
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] The final problem on the optimality of the general theory for nonlinear wave equations and related topics2014

    • Author(s)
      H. Takamura
    • Organizer
      SEOUL ICM 2014, Section10:Partial Differential Equations
    • Place of Presentation
      COEX(韓国京城市)
    • Year and Date
      2014-08-18
    • Related Report
      2014 Annual Research Report
  • [Presentation] 公立はこだて未来大学における数学教育と福原賞受賞研究について2014

    • Author(s)
      高村博之
    • Organizer
      第90回数学教育実践研究会 兼 第20回夏季セミナー
    • Place of Presentation
      北海道小樽桜陽高等学校(北海道小樽市)
    • Year and Date
      2014-08-09
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] Global existence for semilinear wave equations including the blow-up term in four space dimensions2014

    • Author(s)
      H. Takamura
    • Organizer
      The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 60; New Developments in Qualitative Behavior of Evolutionary PDEs
    • Place of Presentation
      マドリード自治大学(スペイン共和国マドリード市)
    • Year and Date
      2014-07-11
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] 非線形波動方程式に対する一般論の完成とその後の発展

    • Author(s)
      高村博之
    • Organizer
      2014年度第1回数理科学談話会
    • Place of Presentation
      室蘭工業大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 高次元正値球対象解と半線形問題への応用

    • Author(s)
      高村博之
    • Organizer
      第29回松山キャンプ
    • Place of Presentation
      山口大学
    • Related Report
      2013 Research-status Report
  • [Presentation] 高次元非線形波動方程式の古典解の解析(特別講演)

    • Author(s)
      高村博之
    • Organizer
      太宰府偏微分方程式研究集会
    • Place of Presentation
      北九州情報大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Blow-up of solutions to semi linear wave equations with non-zero initial data

    • Author(s)
      M.Rammaha, 高村博之, 上坂洋司, 若狭恭平
    • Organizer
      2013日本数学会秋期総合分科会函数方程式論分科会
    • Place of Presentation
      愛媛大学
    • Related Report
      2013 Research-status Report
  • [Presentation] Global existence for semi linear wave equations with the blow-up term in high dimensions

    • Author(s)
      Hiroyuki Takamura
    • Organizer
      The 9th ISAAC Congress
    • Place of Presentation
      ポーランド共和国クラコフ市
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] Positive solutions of high dimensional wave equations with non-zero data

    • Author(s)
      Hiroyuki Takamura
    • Organizer
      Workshop nonlinear PDE
    • Place of Presentation
      イタリア共和国ピサ市
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 空間4次元で2次の非線形項をもつ波動方程式の解析

    • Author(s)
      高村博之
    • Organizer
      解析セミナー
    • Place of Presentation
      神戸大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 空間4次元で2次の非線形項をもつ波動方程式の解析

    • Author(s)
      高村博之
    • Organizer
      波動セミナー
    • Place of Presentation
      北海道大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] 空間4次元で2次の非線形項をもつ波動方程式の解析

    • Author(s)
      高村博之
    • Organizer
      第111回神楽坂解析セミナー
    • Place of Presentation
      東京理科大学
    • Related Report
      2013 Research-status Report
    • Invited
  • [Presentation] General theory for nonlinear wave equations and its optimality

    • Author(s)
      H.Takamura
    • Organizer
      麗水学院理学院談話会
    • Place of Presentation
      中国浙江省麗水市
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 単独非線形波動方程式の初期値問題に対する一般論とその最適性最終問題

    • Author(s)
      高村博之
    • Organizer
      日本数学会2013年度年会函数方程式論分科会特別講演
    • Place of Presentation
      京都大学
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 空間4次元で2次の非線形項をもつ波動方程式に関する消散構造の一例

    • Author(s)
      高村博之, 若狭恭平
    • Organizer
      2012年度日本数学会年会函数方程式論分科会
    • Place of Presentation
      京都大学
    • Related Report
      2012 Research-status Report
  • [Presentation] 空間4次元で2次の爆発項を含む半線形波動方程式に対する時間大域解の存在

    • Author(s)
      高村博之
    • Organizer
      松山解析セミナー2013
    • Place of Presentation
      愛媛大学理学部
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 空間4次元で2次の半線形波動方程式に対する消散構造の解析

    • Author(s)
      高村博之
    • Organizer
      第28回松山キャンプ
    • Place of Presentation
      山口大学理学部
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Global existence for semiline wave equations with the blow-up term in high dimensions

    • Author(s)
      H.Takamura
    • Organizer
      Taiwan-Japan Joint Conference on PDE and Analysis
    • Place of Presentation
      国立台湾大学(台湾)
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 非線形波動方程式に対する一般論の終結と共同研究者育成に関する話題

    • Author(s)
      高村博之
    • Organizer
      第1回釧路解析セミナー
    • Place of Presentation
      北海道教育大学釧路校
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 空間4次元で2次の非単項かつ不定符号の非線形項をもつ波動方程式の解の最大存在時間

    • Author(s)
      高村博之, 若狭恭平
    • Organizer
      2012年度日本数学会秋季総合分科会函数方程式論分科会
    • Place of Presentation
      九州大学
    • Related Report
      2012 Research-status Report
  • [Presentation] Almost global existence for semilinear wave equations with the critical exponent in high dimensions

    • Author(s)
      H.Takamura
    • Organizer
      第37回偏微分方程式札幌シンポジウム
    • Place of Presentation
      北海道大学理学部
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] The final problem on the optimality of the general theory for nonlinear wave equations and related topics

    • Author(s)
      H.Takamura
    • Organizer
      Special session 41, "New Developments in Qualitative Behavior of Evolutionary PDEs" in 9th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Orlando, Florida, USA
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Almost global existence for semilinear wave equations with the critical exponent in high dimensions

    • Author(s)
      高村博之
    • Organizer
      RIMS研究集会「幾何学的偏微分方程式に対する保存則と正則性特異性の研究」
    • Place of Presentation
      京都大学数理解析研究所
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] 高次元波動方程式の正値解と非線形一般論の最適性最終問題

    • Author(s)
      高村博之
    • Organizer
      第542回応用解析研究会
    • Place of Presentation
      早稲田大学理工学部
    • Related Report
      2012 Research-status Report
    • Invited

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Published: 2013-05-31   Modified: 2019-07-29  

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