• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2007 Fiscal Year Final Research Report Summary

Integrated Study for Nonlinear Evolution Equations and Nonlinear Elliptic Equations

Research Project

Project/Area Number 16340043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionWaseda University

Principal Investigator

OTANI Mitsuharu  Waseda University, Faculty of Science and Engineering, Professor (30119656)

Co-Investigator(Kenkyū-buntansha) YAMADA Yoshio  Waseda University, Faculty of Science and Engineering, Professor (20111825)
TANAKA Kazunaga  Waseda University, Faculty of Science and Engineering, Professor (20188288)
ISHII Hitoshi  Waseda University, Faculty of Education and Integrated Arts and Sciences, Professor (70102887)
KENMOCHI Nobuyuki  Chiba University, Faculty of Education, Professor (00033887)
OZAWA Tohru  Hokkaido University, Graduate School of Science, Professor (70204196)
Project Period (FY) 2004 – 2007
KeywordsNonlinear Evolution Equation / Nonlinear Elliptic Equation / Nonlinear PDE / Method of Variation / Subdifferential Operator
Research Abstract

(i) L^∞-energy Method, developed in this research, is applied to the nonlinear parabolic equations with nonlinear terms involving the time derivative to show the existence of the unique local solution. The verification for the uniqueness was difficult for the existing methods because of the lack of regularity. However this method makes it possible by assuring the high regularity of solutions. Furthermore this method turns out to be very effective also for nonlinear parabolic systems for chemotaxis and systems with the hysteresis effect by the fact that it can assure the existence an uniqueness of solution under much weaker conditions than ever
(ii) The infinite dimensional global attractor is constructed in L^2, which attracts all orbits for the initial boundary value problem for the quasi-linear parabolic equation governed by the p-Laplacian. The infinite dimensional global attractor is never observed for the semilinear parabolic equations, so this very new observation seems to be very … More important. On the other hand, the existence of the exponential attractor with finite fractal dimension , which attracts all orbits starting from some special class of initial data exponentially, is shown for some special quasilinear parabolic equations involving Laplacian and p-Laplacian., whence follows the finite dimensionality of the global attractor. These observations suggest that in contrast with semilinear equations, there should exist some structure in quasilinear parabolic equations which controls the finite-dimensionality and infinite-dimensionality of global attractors, which gives a very interesting future object ton study.
iii) It is shown that for Cauchy problem and periodic problem for the abstract evolution equation governed by time-dependent subdifferential operators, if the sequence of approximating subdifferential operators converges to the original one, then the corresponding approximating solutions converge to the solution of the original equation. As for the periodic problem, it is very meaningful to give an affirmative answer to the open problem left long. Less

  • Research Products

    (10 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (6 results) (of which Peer Reviewed: 3 results) Presentation (2 results) Book (2 results)

  • [Journal Article] Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space2008

    • Author(s)
      H.Ishii
    • Journal Title

      Ann.Inst.H.Poincare Anal.Non Lineaire 25

      Pages: 231-266

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space2008

    • Author(s)
      H. Ishii
    • Journal Title

      Ann. Inst. H. Poincare Anal. Non Linearite Vol.25

      Pages: 231-266

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation2006

    • Author(s)
      P.Felmer, S.Martinez and K.Tanaka
    • Journal Title

      Arch.Rat.Mech.Anal. 182

      Pages: 333-366

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] High frequency solutions for singularly perturbed 1D nonlinear Schrodinger equation2006

    • Author(s)
      P. Felmer, S. Martinez, K. Tanaka
    • Journal Title

      Arch. Rat. Mech. Anal. Vol.182

      Pages: 333-366

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] The principle of symmetric criticality for non-differentiable mappings2004

    • Author(s)
      J.Kobayashi and M.Otani
    • Journal Title

      Journal of Functional Analysis 124

      Pages: 428-449

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] The principle of symmetric criticality for non-differentiable mappings

    • Author(s)
      J. Kobayashi, M. Otani
    • Journal Title

      Journal of Functional Analysis vol.214

      Pages: 428-449(20004)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] L∞-energy method-its basic tools and applications-2008

    • Author(s)
      Mitsuharu Otani
    • Organizer
      Mathematical Society of Japan Invited Special Lecture
    • Place of Presentation
      Tohoku University
    • Year and Date
      2008-09-23
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] L∞-エネルギー法について-その基礎と応用-2007

    • Author(s)
      大谷 光春
    • Organizer
      日本数学会 企画特別講演
    • Place of Presentation
      東北大学
    • Year and Date
      2007-09-23
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Mathematical Approach to Nonlinear Phenomena:Modelling, Analysis and Simulations2005

    • Author(s)
      T.Aiki, N.kenmochi, M.Niezgodka and M.Otani
    • Total Pages
      327
    • Publisher
      Gakkotosho Co.,Ltd.
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Mathematical Approach to Nonlinear Phenomena : Modelling, Analysis and Simulations2005

    • Author(s)
      T. Aiki, N. kenmochi, M. Niezgodka, M. Otani
    • Publisher
      Gakkotosho Co., Ltd.
    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2010-02-04  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi