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Exterior problem for nonlinear wave equations

Research Project

Project/Area Number 13440049
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKYUSHU UNIVERSITY

Principal Investigator

NAKAO Mitsuhiro  Kyushu Univetrsity, Graduate School of Mathematics, Professor, 大学院・数理学研究院, 教授 (10037278)

Co-Investigator(Kenkyū-buntansha) KAWASHIMA Syuichi  Kyushu University, Graduate School of Mathematics, Professer, 大学院・数理学研究院, 教授 (70144631)
SHIBATA Yoshihiro  Waseda University, Facultry of Science and Technology, Professor, 理工学部, 教授 (50114088)
OGAWA Takayoshi  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (20224107)
Project Period (FY) 2001 – 2004
Project Status Completed (Fiscal Year 2004)
Budget Amount *help
¥11,400,000 (Direct Cost: ¥11,400,000)
Fiscal Year 2004: ¥2,700,000 (Direct Cost: ¥2,700,000)
Fiscal Year 2003: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2002: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2001: ¥3,100,000 (Direct Cost: ¥3,100,000)
KeywordsNonlinear wave eqations / Exterior problem / Energy decay / Global solutions / 局在非線形的摩擦項 / 波動方程式 / 周期解 / 局在的摩擦項 / 大域解
Research Abstract

The main purpose of this research is concerned with the exterior problem for the quasi-linear wave equations. For this problem we have been successful in proving the global existence of smooth solutions under the effect of localized dissipation. We have achieved the results through two ways; one is based on the local energy decay and L^p estimates of solutions for linear equation, and the other one is the method to utilize total energy decay for the llinearized equation. Both ways are intended to make the effects of dissipation as weaker as possible, but, we have made no geometrical conditions on the shape of the boundary.
Concerning another problem on the energy decay for the equation with nonlinear dissipations we introduced anew concept ‘Half linear' and has been successful in deriving very delicate decay estimates of energy and applied them to the existence of global solutions for the equations with a nonlinear source term.
As related problems we have considered the existence and stability of periodic solutions for the nonlinear wave equations in bounded domains with some nonlinear localized dissipations. Further, we have considered the Kirchhoff type nonlinear wave equations in exterior domains. Under a nonlinear dissipations we have proved various results on global solutions. For the wave equation in exterior domains with a Neumann type boundary dissipation we have derived a new energy decay estimate.
Investigator Kawashima has derived many interesting results concerning Boltzman equations and hyperbolic conservation equations. Investigator Shibata has derived by the method of spectral analysis, many interesting results concerning the exterior problem for the compressive Navier-Stokes equations. Investigator Ogawa has proved precise estimates of solutions concerning behaviors and regularities of solutions for the nonlinear wave equations, nonlinear Shroadinger equations and some harmonic evolution equation.

Report

(5 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (26 results)

All 2005 2004 2003 2002 2001 Other

All Journal Article (15 results) Publications (11 results)

  • [Journal Article] Energy decay for the wave equation in exterior domains with a localized and a boundary dissipation.2005

    • Author(s)
      J.J.Bae, M.Nakao
    • Journal Title

      Nath.Nachr. 278

      Pages: 1-13

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains.2004

    • Author(s)
      J.J.Bae, M.Nakao
    • Journal Title

      Discrete Contin.Dyn.Syst. 11

      Pages: 731-743

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Global existence for the Kirchhoff type quasilinear wave equation in exterior domains with a halflinear dissipation.2004

    • Author(s)
      M.Nakao
    • Journal Title

      Kyushu J.Math. 58

      Pages: 373-391

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Global and periodic solutions for the nonlinear wave equations with some nonlinear localized dissipations.2003

    • Author(s)
      M.Nakao
    • Journal Title

      J.Differential Equations 190

      Pages: 81-107

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Global existence of the smooth solutions to the initial boundary value problem for the quasilinear wave equations in exterior domains.2003

    • Author(s)
      M.Nakao
    • Journal Title

      J.Math.Soc.Japan 35

      Pages: 765-795

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Global and periodic solutions for the nonlinear wave equations with some nonlinear localized dissipations2003

    • Author(s)
      M.Nakao
    • Journal Title

      J.Differential Equations 190

      Pages: 81-107

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Global existence of the smooth solutions to the initial boundary value problem for the quasilinear wave equations in exterior domains2003

    • Author(s)
      M.Nakao
    • Journal Title

      J.Math.Soc.Japan 35

      Pages: 765-795

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Existence of global solutions to the Cauchy problem for the Kirchoff type quasilinear wave equations with a weakly nonlinear dissipation.2002

    • Author(s)
      M.Nakao
    • Journal Title

      Funk.Ekvac. 45

      Pages: 387-395

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Existence of global solutions to the Cauchy problem for the Kirch-hoff type quasilinear wave equations with a weakly nonlinear dissipation2002

    • Author(s)
      M.Nakao, J.J.Bae
    • Journal Title

      Funk.Ekvac. 45

      Pages: 387-395

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations.2001

    • Author(s)
      M.Nakao
    • Journal Title

      Math.Z. 238

      Pages: 781-797

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] L^p estimates for the wave equation and global existence for the semilinear wave equations in exterior domains.2001

    • Author(s)
      M.Nakao
    • Journal Title

      Math.Ann. 320

      Pages: 11-31

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The contact problem in thermoviscoelastic materials.2001

    • Author(s)
      M.Nakao
    • Journal Title

      J.Math.Anal.Appl. 264

      Pages: 522-545

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations2001

    • Author(s)
      M.Nakao
    • Journal Title

      Math.Z. 238

      Pages: 781-797

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] L^p$estimates for the wave equation and global existence for the semilinear wave equations in exterior domains2001

    • Author(s)
      M.Nakao
    • Journal Title

      Math.Ann. 320

      Pages: 11-31

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] The contact problem in thermoviscoelastic materials2001

    • Author(s)
      M.Nakao, J.M.Rivera
    • Journal Title

      J.Math.Anal.Appl. 264

      Pages: 522-545

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Publications] Mitsuhiro Nakao: "Global and periodic solutions for the nonlinear wave equations with some nonlinear localized dissinations"J.Differential Equations. 190. 81-107 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "On global smooth solutions to the initial boundary value problem for the quasilinear wave equation in the exterior domains"J.Mathematical Society of Japan. 35. 765-795 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Mitsuhiro Nakao, Jeong.J.Bae: "Existence of global solutions to the Kirchhoff type equations with some localized dissipations"Adv.Math.Sci.Appl.. 13. 165-177 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Mitsuhiro Nakao, Il.H.Jung: "Energy decay for the wave equation with a half linear dissipation in exterior domains"Differential and Integral Equations. 16. 927-948 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Jeong J.Bae, Mitsuhiro Nakao: "Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains"Disc.Contin.Dyn.Syst.. (To appear).

    • Related Report
      2003 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "Existence of global solutions to the Cauchy problem of Kirchhoff type quasilinear wave equation with weakly nonlinear dissipation"Funkcialaj Ekvacioj. 45・3. 387-395 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "On global smooth solutions to the initial boundary value problem for the quasilinear wave equation in the exterior domains"J. Mathematical Society Japan. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "Global and periodic solutions for the nonlinear wave equation with a localized nonlinear dissipation"J. Differential Equations. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "The contact problem in thermoviscoelastic materials"Journal of Mathematical Analysis and Applications. 264. 522-545 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "Energy deay for the linear and semilinear wave equations in exterior domains with some localized dissipations"Mathematische Zeitschrift. 238. 781-797 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mitsuhiro Nakao: "Lp estimates for the linear wave equation and global existen ce for semilinear wave equations in exterior domains"Mathematishe Annalen. 320. 11-31 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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