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2010 Fiscal Year Final Research Report

New development of the qualitative theory of nonlinear parabolic and elliptic equations

Research Project

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Project/Area Number 19204014
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTokyo Institute of Technology (2010)
Tohoku University (2007-2009)

Principal Investigator

YANAGIDA Eiji  東京工業大学, 大学院・理工学研究科, 教授 (80174548)

Co-Investigator(Kenkyū-buntansha) IZUMI Takagi  東北大学, 大学院・理学研究科, 教授 (40154744)
NAITO Yuki  愛媛大学, 大学院・理工学研究科, 教授 (10231458)
OGAWA Takayoshi  東北大学, 大学院・理学研究科, 教授 (20224107)
EI Shin-ichiro  九州大学, 大学院・数理学研究院, 教授 (30201362)
ISHIGE Kazuhiro  早稲田大学, 理工学術院, 教授 (90272020)
TANAKA Kazunaga  早稲田大学, 理工学術院, 教授 (20188288)
NINONMIYA Hiro  龍谷大学, 理工学部, 准教授 (90251610)
TONEGAWA Yoshihiro  北海道大学, 大学院・理学研究科, 教授 (80296748)
Project Period (FY) 2007 – 2010
Keywords関数方程式の大域理論
Research Abstract

We carried out the investigation about the structure of solutions of nonlinear parabolic and elliptic equations. Our main results are as follows : Next, we studied the existence and uniqueness of solutions with moving singularities for a nonlinear parabolic partial differential equation. We also showed that there exists a solution with a moving singularity that changes its type suddenly., and made clear the asymptotic behavior of singular solutions that converges to a singular steady state. We also studied a chemotaxis system, and made clear the structure of self-similar solutions that blows up by concentrating to a point in finite time.
For a reaction-diffusion system, which is called a Gierer-Meinhardt system, we studied the mathematical structure of pattern formation, and also made clear the behavior of time-dependent solutions.

  • Research Products

    (19 results)

All 2010 2009 2008 2007 Other

All Journal Article (16 results) (of which Peer Reviewed: 15 results) Presentation (2 results) Remarks (1 results)

  • [Journal Article] Forwardself-similar solution with a movingsingularity for a semilinear parabolic equation2010

    • Author(s)
      S. Sato and E. Yanagida
    • Journal Title

      Disc. Cont. Dyn. Systems

      Volume: 26 Pages: 313-331

    • Peer Reviewed
  • [Journal Article] Speeds of Traveling Waves in Some Integro-Differential Equations Arising from Neuronal Networks2010

    • Author(s)
      E. Yanagida and L. Zhang
    • Journal Title

      Japan J Indust. App. Math

      Volume: 27 Pages: 347-373

    • Peer Reviewed
  • [Journal Article] Convergence of anisotropically decaying solutions fora supercritical semilinear heat equation2009

    • Author(s)
      P. Polacik and E. Yanagida
    • Journal Title

      J Dyn. Diff. Eqs

      Volume: 21 Pages: 329-341

    • Peer Reviewed
  • [Journal Article] Slow convergence to zero for a parabolic equation with supercritical nonlinearity2008

    • Author(s)
      M. Fila, M. Winkler and E. Yanagida
    • Journal Title

      Math Annal

      Volume: 340 Pages: 477-496

    • Peer Reviewed
  • [Journal Article] Convergence to self similar solutions fora semilinear parabolic equation2008

    • Author(s)
      M. Fila, M. Winkler and E. Yanagida
    • Journal Title

      Disc Cont. Dyn. Systems

      Volume: 21 Pages: 703-716

    • Peer Reviewed
  • [Journal Article] Sharpestimate of the convergence rate for asemilinear parabolic equation with supercritical nonlinearity2008

    • Author(s)
      M. Hoshino and E. Yanagida
    • Journal Title

      NonlinearAnalysis TMA

      Volume: 69 Pages: 3136-3152

    • Peer Reviewed
  • [Journal Article] Eigenfunctions of the adjointoperator associated with a pulse solution of some reaction-diffusion systems2008

    • Author(s)
      S.-I. Ei, H. Ikeda, K. Ikeda and E. Yanagida
    • Journal Title

      Bull Inst. Math. Acad. Sin

      Volume: 3 Pages: 603-666

    • Peer Reviewed
  • [Journal Article] Representation formula for the critical points of theTadjbakhsh-Odeh functional and its application2008

    • Author(s)
      K. Watanabe, I. Takagi
    • Journal Title

      Japan J Indust. Appl. Math

      Volume: 25 Pages: 331-372

    • Peer Reviewed
  • [Journal Article] Self-similar solutions for asemilinear heat equation with criticalSobolev exponent2008

    • Author(s)
      Y. Naito
    • Journal Title

      Indiana Univ. Math. J

      Volume: 57 Pages: 1283-1315

  • [Journal Article] The driftdiffusion system in two-dimensionalcritical Hardy space.2008

    • Author(s)
      T. Ogawa and S. Shimizu
    • Journal Title

      J Funct. Anal

      Volume: 255 Pages: 1107-1138

    • Peer Reviewed
  • [Journal Article] Convergence of phase-field approximations to the Gibbs-Thomson law.2008

    • Author(s)
      M Roger and Y. Tonegawa
    • Journal Title

      Calc. Var Partial Differential Equations

      Volume: 32 Pages: 111-136

    • Peer Reviewed
  • [Journal Article] Grow-up of solutions of asemilinear parabolic equation with acritical exponent2007

    • Author(s)
      M. Fila, J. R. King, M. Winkler and E. Yanagida
    • Journal Title

      Adv Diff. Eqs

      Volume: 12 Pages: 1-26

    • Peer Reviewed
  • [Journal Article] Irregular behavior of solutions for Fisher's equation2007

    • Author(s)
      E. Yanagida
    • Journal Title

      J Dyn. Diff. Eqs

      Volume: 19 Pages: 895-914

    • Peer Reviewed
  • [Journal Article] On the role of the source terms in an activator-inhibitor system proposed by Gierer and Meinhardt. Asymptotic analysis and singularities-elliptic and parabolic PDEsand related problems2007

    • Author(s)
      K. Suzuki, I. Takagi
    • Journal Title

      Pure Math

      Volume: 47-2 Pages: 749-766

    • Peer Reviewed
  • [Journal Article] Movement of hot spots on the exterior domain of a ball under the Dirichlet boundary condition2007

    • Author(s)
      K. Ishige
    • Journal Title

      Adv. Differential Equations

      Volume: 12 Pages: 1135-1166

    • Peer Reviewed
  • [Journal Article] Decay rates of the derivatives of the solutions of the heat equations in the exterior domain ofa ball2007

    • Author(s)
      K. Ishige, Y. Kabeya
    • Journal Title

      J Math. Soc. Japan

      Volume: 59 Pages: 861-898

    • Peer Reviewed
  • [Presentation] Global and non-globalsolutions with moving singularities fora semilinear parabolic equation2009

    • Author(s)
      Eiji Yanagida
    • Organizer
      The 7thEast-Asia conference on PEDs
    • Place of Presentation
      Chinese University of Hong Kong and the City University of Hong Kong
    • Year and Date
      20090000
  • [Presentation] Solutions with moving singularities for a semilinear parabolic equations2008

    • Author(s)
      Eiji Yanagida
    • Organizer
      7th AIMS international conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      University of Texas at Arlington, USA
    • Year and Date
      20081821
  • [Remarks]

    • URL

      http://www.math.titech.ac.jp/~yanagida/index.html

URL: 

Published: 2013-07-31   Modified: 2014-12-02  

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