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Global existence and the asymptotic behavior for partial diffierential equations concerning nonlinear waves

Research Project

Project/Area Number 26400168
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionOsaka University (2016-2017)
Wakayama University (2014-2015)

Principal Investigator

Katayama Soichiro  大阪大学, 理学研究科, 教授 (70283942)

Project Period (FY) 2014-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords非線形波動方程式 / 大域解 / 零条件 / 弱零条件 / 漸近挙動 / 初期値問題 / クライン・ゴルドン方程式 / 非線形波動 / エネルギー減衰
Outline of Final Research Achievements

We studied sufficient conditions for the existence of global solutions (solutions up to the arbitrary time) to the Cauchy problem for systems of nonlinear wave equations, or for some related systems, with small initial data. Concerning sufficient conditions weaker than the well-known null condition (such weaker conditions are called the "weak" null conditions), we unified the two known "weak" null conditions, and proved the global existence under this unified condition. Some formula to give the asymptotic behavior for global solutiuons was obtained. We also improved the global existence theorem by Alinhac for wave equations in two space dimensions. Concerning the systems of nolinear wave and Klein-Gordon equations, we proved the global existence of solutions under a weaker condition than before, in the case that initial data vanish outside a bounded region.

Report

(5 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (17 results)

All 2018 2017 2016 2015 2014

All Journal Article (3 results) (of which Peer Reviewed: 3 results) Presentation (11 results) (of which Int'l Joint Research: 7 results,  Invited: 11 results) Book (1 results) Funded Workshop (2 results)

  • [Journal Article] Global existence for systems of nonlinear wave and Klein-Gordon equations with compactly supported initial data2018

    • Author(s)
      Soichiro Katayama
    • Journal Title

      Communications on Pure and Applied Analysis

      Volume: 17 Issue: 4 Pages: 1479-1497

    • DOI

      10.3934/cpaa.2018071

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Remarks on the asymptotic behavior of global solutions to systems of semilinear wave equations2018

    • Author(s)
      Soichiro Katayama
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: -

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Global existence for a class of system of nonlinear wave and Klein-Gordon equations2018

    • Author(s)
      Soichiro Katayama
    • Journal Title

      RIMS Kokyuroku Bessatsu (数理解析研究所講究録別冊)

      Volume: -

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Presentation] Global existence and the asymptotic behavior for systems of semilinear wave equations2017

    • Author(s)
      Soichiro Katayama
    • Organizer
      Workshop on Nonlinear Wave Equations, Fudan University
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Global existence for a class of system of nonlinear wave and Klein-Gordon equations in 3D2017

    • Author(s)
      Soichiro Katayama
    • Organizer
      調和解析と非線形偏微分方程式 (於: 京都大学数理解析研究所)
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Global existence for a class of system of nonlinear wave and Klein-Gorodn equations2017

    • Author(s)
      片山 聡一郎
    • Organizer
      第13回非線型の諸問題 (於: 鹿児島市町村自治会館)
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Decay of solutions to nonlinear dissipative wave equations in one space dimension2017

    • Author(s)
      片山 聡一郎
    • Organizer
      広島微分方程式研究会 (於: 広島大学)
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Decay of solutions of wave equations with nonlinear dissipative terms2017

    • Author(s)
      Soichiro Katayama
    • Organizer
      研究集会「Critical Exponents and Nonlinear Evolution Equation」
    • Place of Presentation
      東京理科大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Global existence for systems of nonlinear wave and Klein-Gordon equations with compactly supported data2017

    • Author(s)
      Soichiro Katayama
    • Organizer
      Zhejiang-Tohoku International Workshop "Nonlinear Partial Differential Equations 2017"
    • Place of Presentation
      東北大学
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 非線形波動方程式系の大域解の存在と漸近挙動2017

    • Author(s)
      片山 聡一郎
    • Organizer
      日本数学会2017年度年会(函数方程式論分科会特別講演)
    • Place of Presentation
      首都大学東京
    • Related Report
      2016 Research-status Report
    • Invited
  • [Presentation] Asymptotic behavior of solutions for systems of semilinear wave equations2016

    • Author(s)
      Soichiro Katayama
    • Organizer
      The 23rd Machikaneyama Seminar on PDEs
    • Place of Presentation
      大阪大学 理学部
    • Year and Date
      2016-03-08
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behavior of solutions for systems of semilinear wave equations2016

    • Author(s)
      片山 聡一郎
    • Organizer
      第33回九州における偏微分方程式研究集会
    • Place of Presentation
      九州大学 西新プラザ
    • Year and Date
      2016-01-29
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behavior for systems of semilinear wave equations violating the null condition2015

    • Author(s)
      Soichiro Katayama
    • Organizer
      Workshop on Partial Differential Equations and Numerical Analysis
    • Place of Presentation
      延辺大学 (中国)
    • Year and Date
      2015-07-07
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behavior for small solutions to systems of semilinear wave equations2014

    • Author(s)
      Soichiro Katayama
    • Organizer
      International Conference: Asymptotic Analysis for Nonlinear Dispersive and Wave Equations
    • Place of Presentation
      大阪大学
    • Year and Date
      2014-09-11
    • Related Report
      2014 Research-status Report
    • Invited
  • [Book] Global Solutions and the Asymptotic Behavior for Nonlinear Wave Equations with Small Initial Data (MSJ Memoirs Volume 36)2017

    • Author(s)
      Soichiro Katayama
    • Total Pages
      298
    • Publisher
      Mathematical Society of Japan (日本数学会)
    • ISBN
      9784864970549
    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Linear and Nonlinear Waves No. 152017

    • Related Report
      2017 Annual Research Report
  • [Funded Workshop] Linear and Nonlinear Waves, No.142016

    • Place of Presentation
      ピアザ淡海 滋賀県県立交流センター
    • Year and Date
      2016-11-02
    • Related Report
      2016 Research-status Report

URL: 

Published: 2014-04-04   Modified: 2019-03-29  

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