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2013 Fiscal Year Final Research Report

Global existence and asymptotic behavior for systems of nonlinear hyperbolic equations

Research Project

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Project/Area Number 23540241
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Global analysis
Research InstitutionWakayama University

Principal Investigator

KATAYAMA Soichiro  和歌山大学, 教育学部, 教授 (70283942)

Project Period (FY) 2011 – 2013
Keywords波動方程式 / 非線形 / 零条件 / 大域解 / 漸近挙動 / 弱零条件
Research Abstract

We studied sufficient conditions for global exitence of solutions and asymptotic behavior of global solutions to the Cauchy problem (or the exterior problem) for (systems of) nonlinear hyperbolic equations. For complex-valued (or systems of) semilinear wave equations in two space dimensions, we obtained sufficient conditions, which are weaker than the so-called null condition, for small data global existence, and also obtained sufficient conditions for the energy decay. For the exterior problem for semilinear or quasi-linear wave equations in two and three spece dimensions, we derived a lower bound for the lifespan of classical solutions when the null condition is not satisfied. We also clarified the null conditions for massless Dirac equations and systems of nonlinear Schroedinger equations, and proved that the global solutions are asymptotically free.

  • Research Products

    (11 results)

All 2014 2013 2012 Other

All Journal Article (7 results) (of which Peer Reviewed: 7 results) Presentation (4 results) (of which Invited: 2 results)

  • [Journal Article] A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D2014

    • Author(s)
      Soichiro Katayama, Chunhua Li and Hideaki Sunagawa
    • Journal Title

      Differential and Integral Equations

      Volume: 27, No.3-4 Pages: 301-312

    • URL

      http://projecteuclid.org/euclid.die/1391091368

    • Peer Reviewed
  • [Journal Article] Global existence for quadratically perturbed massless Dirac equations under the null condition2014

    • Author(s)
      Soichiro Katayama and Hideo Kubo
    • Journal Title

      Fourier Analysis: Pseudo-Differential Operators, Time-Frequency Analysis and Partial Differential Equations (M. Ruzhansky and V. Turunen, eds.), Birkhäuser

      Pages: 253- 262

    • Peer Reviewed
  • [Journal Article] Almost global existence for exterior Neumann problems of semilinear wave equations in 2D2013

    • Author(s)
      Soichiro Katayama, Hideo Kubo and Sandra Lucente
    • Journal Title

      Communications on Pure and Applied Analysis

      Volume: 12, No.6 Pages: 2331-2360

    • DOI

      10.3934/cpaa.2013.12.2331

    • Peer Reviewed
  • [Journal Article] Asymptotic behavior for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions2013

    • Author(s)
      Soichiro Katayama
    • Journal Title

      Journal of Differential Equations

      Volume: 255, No.1 Pages: 120-150

    • DOI

      10.1016/j.jde.2013.04.003

    • Peer Reviewed
  • [Journal Article] Lower bound of the lifespan of solutions to semilinear wave equations in an exterior domain2013

    • Author(s)
      Soichiro Katayama and Hideo Kubo
    • Journal Title

      Journal of Hyperbolic Differential Equations

      Volume: 10, No.2 Pages: 199-234

    • DOI

      10.1142/S0219891613500094

    • Peer Reviewed
  • [Journal Article] The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions2012

    • Author(s)
      Soichiro Katayama, Daisuke Murotani and Hideaki Sunagawa
    • Journal Title

      Journal of Evolution Equations

      Volume: 12, No.4 Pages: 891-916

    • DOI

      10.1007/s00028-012-0160-4

    • Peer Reviewed
  • [Journal Article] Null structure in a system of quadratic derivative Schrödinger equations

    • Author(s)
      Masahiro Ikeda, Soichiro Katayama and Hideaki Sunagawa
    • Journal Title

      Annales Henri Poincaré

      Volume: (印刷中)

    • DOI

      10.1007/s00023-014-0316-6

    • Peer Reviewed
  • [Presentation] Semilinear hyperbolic systems violating the null condition2013

    • Author(s)
      Soichiro Katayama
    • Organizer
      Mexico-Japan Joint Meeting on PDE at Morelia
    • Place of Presentation
      メキシコ自治大学モレリアキャンパス
    • Year and Date
      2013-09-05
  • [Presentation] A semilinear hyperbolic system violating the null condition2013

    • Author(s)
      片山 聡一郎
    • Organizer
      第30回九州における偏微分方程式研究集会
    • Place of Presentation
      福岡大学
    • Year and Date
      2013-01-31
    • Invited
  • [Presentation] A semilinear hyperbolic system violating the null condition2012

    • Author(s)
      Soichiro Katayama
    • Organizer
      China-Japan Joint Meeting on PDE at Yanji
    • Place of Presentation
      延辺大学
    • Year and Date
      2012-09-10
  • [Presentation] 異なる伝播速度を持つ非線形波動方程式系の解の漸近挙動2012

    • Author(s)
      片山 聡一郎
    • Organizer
      函館偏微分方程式研究会
    • Place of Presentation
      公立はこだて未来大学
    • Year and Date
      2012-01-28
    • Invited

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Published: 2015-07-16  

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