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2008 Fiscal Year Final Research Report

Almost periodic oscillations of linear and nonlinear hyperbolic equations

Research Project

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Project/Area Number 18540220
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTokai University

Principal Investigator

YAMAGUCHI Masaru  Tokai University, 理学部数学科, 教授 (10056252)

Co-Investigator(Kenkyū-buntansha) MATSUYAMA Tokio  東海大学, 理学部, 教授 (70249712)
NARAZAKI Takashi  東海大学, 理学部, 教授 (70119692)
Co-Investigator(Renkei-kenkyūsha) TANAKA Minoru  東海大学, 理学部, 教授 (10112773)
Project Period (FY) 2006 – 2008
Keywords双曲型方程式 / 概周期解 / Diophantine 条件
Research Abstract

双曲型偏微分方程式の典型的で物理的にも重要な方程式である3つの方程式、波動方程式・Klein-Gordon方程式・吊り下げられた弦の方程式について、線形・非線形双方の場合について、主として境界値問題の周期解の存在に関する研究を行い、上記3つの方程式が非線形自励系の場合に、境界値問題に対して連続濃度のパラメーターに依存する無限個の周期解の存在に関する結果を得た。その際、線型方程式の周期解に関する正則性評価と周期と主要部の楕円型作用素のスペクトルに対する数論による評価と非線形方程式のLyapunov-Schmidt分解が基本的に用いられた。このとき、解の正則性について非線形項の微分可能性とスペクトルと周期に関するDiophantine不等式におけるDiophantine指数との整合性のある結果を得た。
また、吊り下げられた弦の方程式が線形減衰項をもつ場合、周期的な非線形をもつときには、局所一意的な周期解をもつことを示した。さらに概周期解について存在とその安定性が証明された(服部仁実氏(準備中))。また、自励非線形方程式に対する初期値境界値問題の時間大域解の存在を証明した。
研究活動として、関連する分野の国際・国内研究集会「発展方程式シンポジウム」を各年度ごとに開催し(本研究期間中に3回)国内外の研究者と活発に情報交換と討論を行い当該分野の研究を推進した。

  • Research Products

    (26 results)

All 2009 2008 2007 2006 Other

All Journal Article (14 results) Presentation (12 results)

  • [Journal Article] (with Wongsawasdi, Jaipong)Global classical solutions of IBVP to nonlinear equation of a suspended string2008

    • Author(s)
      山口勝(Masaru Yamaguchi)
    • Journal Title

      Tokyo J. Math 31, no.2

      Pages: 351-373

  • [Journal Article] Infinitely many periodic solutions of nonlinear equations of suspended strings2008

    • Author(s)
      山口勝(Masaru Yamaguchi)
    • Journal Title

      Funkcial. Ekvac 51, no.2

      Pages: 245-267

  • [Journal Article] Global smooth solutions of IBVP to nonlinear equation of suspended string2008

    • Author(s)
      山口勝(Masaru Yamaguchi)
    • Journal Title

      J. Math. Anal. Appl. 342, no.2

      Pages: 798-815

  • [Journal Article] (with Nagai, Tohru ; Matsukane, Katsuya)Forced oscillations of nonlinear damped equation of suspended string2008

    • Author(s)
      山口勝(Masaru Yamaguchi)
    • Journal Title

      J. Math. Anal. Appl. 342, no.1

      Pages: 89-107

  • [Journal Article] L^p-L^q estimates for wave equations and the Kirchhoff equation2008

    • Author(s)
      松山登喜夫(Tokio Matsuyama)
    • Journal Title

      Osaka J. Math 45, no.2

      Pages: 491-510

  • [Journal Article] (with Nishihara, Kenji)Asymptotic behavior of solutions for the damped wave equation with slowly decaying data2008

    • Author(s)
      楢崎隆(Takashi Narazaki)
    • Journal Title

      J. Math. Anal. Appl. 338, no.2

      Pages: 803-819

  • [Journal Article] (with Wongsawasdi, Jaipong)Global solutions of IBVP to nonlinear equation of suspended string2007

    • Author(s)
      山口勝(Masaru Yamaguchi)
    • Journal Title

      Tokyo J. Math. 30, no.2

      Pages: 543-556

  • [Journal Article] (with Ruzhansky, M.)Dispersion and asymptotic profiles for Kirchhoff equations. Topics in contemporary differential geometry, complex analysis and mathematical physics2007

    • Author(s)
      松山登喜夫(Tokio Matsuyama)
    • Journal Title

      World Sci. Publ., Hackensack, NJ

      Pages: 234-243

  • [Journal Article] (with Sinclair,Robert)The cut locus of a two-shere of revolutions and Toponogov's comparison theorem2007

    • Author(s)
      田中實(Minoru Tanaka)
    • Journal Title

      Tohoku Math. J. 59

  • [Journal Article] Asymptotic profiles for the Kirchhoff equation. Atti Accad. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9)2006

    • Author(s)
      松山登喜夫(Tokio Matsuyama)
    • Journal Title

      Mat. Appl. no.4

      Pages: 377-395

  • [Journal Article] (with Sinclair,Robert)Jacobi's last geometric statement extends to a wider class of Liouville surfaces2006

    • Author(s)
      田中實(Minoru Tanaka)
    • Journal Title

      Math. Computation 75

      Pages: 1779-1808

  • [Journal Article] Existence and regularity of periodic solutions of nonlinear equations of a suspended string

    • Author(s)
      山口勝(Masaru Yamaguchi)
    • Journal Title

      to appear in Funkcialaj Ekvac

  • [Journal Article] (with Reissig, Michael)Stabilization and L^p-L^q decay estimates

    • Author(s)
      松山登喜夫(Tokio Matsuyama)
    • Journal Title

      Asymptot. Anal. 50, no.3-4

      Pages: 239-268

  • [Journal Article] Asymptotic behaviour for wave equation with time-dependent coefficients

    • Author(s)
      松山登喜夫(Tokio Matsuyama)
    • Journal Title

      Ann. Univ. Ferrara Sez. VII Sci. Mat. 52, no.2

      Pages: 383-393

  • [Presentation] Global solutions to Cauchy problem for the weakly coupled system of damped wave equations2009

    • Author(s)
      楢崎隆
    • Organizer
      第16回 応用解析研究会シンポジウム
    • Place of Presentation
      箱根
    • Year and Date
      20090300
  • [Presentation] Periodic oscillations of nonlinear suspended string equations2008

    • Author(s)
      山口勝
    • Organizer
      偏微分方程式の諸問題
    • Place of Presentation
      東海大学
    • Year and Date
      20081000
  • [Presentation] Geometrical optics for hyperbolic equations2008

    • Author(s)
      松山登喜夫
    • Organizer
      Decay and regularity for solutions of differential equations and dynamical systems
    • Place of Presentation
      Cagliari, Italy
    • Year and Date
      20080900
  • [Presentation] Dispersion for the Kirchhoff equations2008

    • Author(s)
      松山登喜夫
    • Organizer
      Isaac Congress 2007
    • Place of Presentation
      Ankara, Turkey
    • Year and Date
      20080800
  • [Presentation] Global smooth solutions of IBVP to nonlinear suspended string equations2008

    • Author(s)
      山口勝
    • Organizer
      7th AIMS International Conference Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Texas, USA
    • Year and Date
      20080500
  • [Presentation] Global solutions to the Cauchy problem for the system of damped wave equations2008

    • Author(s)
      楢崎隆
    • Organizer
      7th AIMS International Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Texas, USA
    • Year and Date
      20080500
  • [Presentation] Scattering for the Kirchhoff equation with nonlinear perturbations2008

    • Author(s)
      松山登喜夫
    • Organizer
      Function spaces and partial differential equations
    • Place of Presentation
      大阪大学
    • Year and Date
      20080200
  • [Presentation] Periodic solutions of 2D nonlinear wave equations in periodic domains2007

    • Author(s)
      山口勝
    • Organizer
      International Conference of Numerical Analysis and Applied Mathematics 2007
    • Place of Presentation
      Corfu, Greece
    • Year and Date
      20070900
  • [Presentation] Periodic solutions of nonlinear equations of suspended string2007

    • Author(s)
      山口勝
    • Organizer
      Conference "Interplay between Peudo-differential Operators and PDEs" 110 Years Anniversary : Fransesco Giacomo Tricomi
    • Place of Presentation
      Torino, Italy
    • Year and Date
      20070100
  • [Presentation] Asymptotic profiles for Kirchhoff equation2007

    • Author(s)
      松山登喜夫
    • Organizer
      第24回九州における偏微分方程式研究集会
    • Place of Presentation
      九州大学た
    • Year and Date
      20070100
  • [Presentation] Global solvability to Kirchhoff equations of higher ordera2007

    • Author(s)
      松山登喜夫
    • Organizer
      Interpaly between pseudo-differential operators and PDEs
    • Place of Presentation
      Torino, Italy
    • Year and Date
      20070100
  • [Presentation] Periodic solutions of nonlinear suspended string equations2006

    • Author(s)
      山口勝
    • Organizer
      International Conference on Nonlinear Evolutionary PDEs
    • Place of Presentation
      Xining, China
    • Year and Date
      20060700

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Published: 2010-06-10   Modified: 2016-04-21  

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