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On the behavior of solutions for nonlinear parabolic or elliptic equations

Research Project

Project/Area Number 13640205
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTOKYO GAKUGEI UNIVERSITY

Principal Investigator

MIZOGUCHI Noriko  Tokyo Gakugei University, Department of Mathematics, Assistant professor, 教育学部, 助教授 (00251570)

Co-Investigator(Kenkyū-buntansha) KUBOTA Yoshihisa  Tokyo Gakugei University, Department of Mathematics, Professor, 教育学部, 教授 (30014715)
YAMADA Akira  Tokyo Gakugei University, Department of Mathematics, Professor, 教育学部, 教授 (60126331)
YANAGIDA Eiji  Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
ITO Ichiro  Tokyo Gakugei University, Department of Mathematics, Assistant professor, 教育学部, 助教授 (60134764)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2001: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordsnonlinear / parabolic / blow-up / global solution / critical exponent / complementary space / stable domain
Research Abstract

It seems that there have been no results on the blow-up of solutions for a semilinear heat equation with power nonlinearity under the Neumann boundary condition in a bounded domain since the well-known result by Giga and Kohn cannot work effectively, which is a quite difference from the Cauchy or the Dirichlet-Cauchy problem. I proved that the blow-up rate and behaviors of solutions are as same as that in the result by Giga and Kohn by a joint work with Kazuhiro Ishige of Nagoya University. Moreover, we showed that the bllow-up occurs only near the maximum points of the second eigenfunctions of the laplacian with the Nuemann boundary condition. This result is closely related to the hot spots conjecture for the heat equation.
On the other hand, the structure of global solutions of the Cauchy problem for tie above equation changes suddenly near some exponent Yanagida obtained the global stability of stationary solutions. As a application, he showed the existence of unbounded global solutions which behave very complicatedly.
Yamada investigated some fundamental properties about the complementary spaces on the linear contraction of de Branges and obtained a simple proof the problem on the extended interpolation in the case of the unit ball due to Takahashi of Nara University.
Kubota studied about the stable domain of automorphism in a Banach space.
Ito tried to apply the above studies to practical information education.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (51 results)

All Other

All Publications (51 results)

  • [Publications] N.Mizoguchi, K.Ishige: "Blow-up behavior for semilinear heat equations with Neumann boundary conditions"J.Differential Integral Equations. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Mizoguchi: "On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity"Math.Z. 239. 215-229 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Mizoguchi, E.Yanagida: "Life span of solutions for a semilinear parabolic problem with small diffusion"J.Math.Anal.Appl.. 261. 350-368 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Mizoguchi: "Asymptotic behavior of zeros of solutions for parabolic equations"J.Differential Equations. 170. 51-67 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Mizoguchi, E.Yanagida: "Life span of solutions with large initial data in a semilinear parabolic equation"Indiana Univ.Math.J.. 50. 591-610 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] P.Polacik, E.Yanagida: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc.Cont.Dyn.Systems. 8. 209-218 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"J.Differential Equations. 179. 311-335 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida: "Standing Pulse Solutions in Reaction-Diffusion Systems with Skew-Gradient Structure"J.Dyn.Diff.Eqs. 4. 189-205 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida: "Stability of nonconstant stationary solutions for reaction-diffusion systems on graphs"Japan.J.Indust.Appl.Math. 18. 25-42 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida, W.M.Ni, I.Takagi: "Stability of least energy patterns of the shadow system for an activator-inhibitor model"Japan.J.Indust.Appl.Math. 18. 259-272 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida, W.M.Ni, P.Polacik: "Monotonicity of stable solutions in shadow systems"Trans.Amer.Math.Soc.. 353. 5057-5069 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida, P.Polacik: "Existence of stable subharmonic solutions for reaction-diffusion equations"J.Differential Equations. 169. 255-280 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A.Yamada: "Ahlfors functions on compact bordered Riemann surfaces"J.Math.Soc.Japan. 53. 261-283 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Kubota: "Stable domains of automorphisms of Banach spaces at fixed points"Bulletin of Tokyo Gakugei University, Sect.IV Mathematics and Natural Sciences. 53. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 伊藤一郎: "中学校における情報教育(東京学芸大学附属中学校での検討課題)167-174"情報処理学会情報教育シンポジウム論文集,IPSJ Symposium Series. 9. (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N. Mizoguchi and K. Ishige: "Blow-up behavior for semilinear heat equations with Neumann boundary conditions"J. Differential Integral Equations. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N. Mizoguchi: "On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity"Math. Z.. 239. 215-229 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N. Mizoguchi and E. Yanagida: "Life span of solutions for a semilinear parabolic problem with small diffusion"J. Math. Anal. Appl.. 261. 350-368 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N. Mizoguchi: "Asymptotic behavior of zeros of solutions for parabolic equations"J. Differential Equations. 170. 51-67 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N. Mizoguchi and E Yanagida: "Life span of solutions with large initial data in a semilinear parabolic equation"Indiana Univ. Math. J.. 50. 591-610 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] P. Polacik and E. Yanagida: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc. Cont. Dyn. Systems.. 8. 209-218 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"J. Diff. Eqs.. 179. 311-335 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida: "Standing Pulse Solutions in Reaction-Diffusion Systems with Skew-Gradient Structure"J. Dyn. Diff. Eqs.. 4. 189-205 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida: "Stability of nonconstant stationary solutions for reaction-diffusion systems on graphs"Japan. J. Indust. Appl. Math.. 18. 25-42 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida, W. M. Ni and I. Takagi: "Stability of least energy patterns of the shadow system for an activator-inhibitor model"Japan J. Indust. Appl. Math.. 18. 259-272 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida, W. M. Ni and P. Polacik: "Monotonicity of stable solutions in shadow Systems"Trans. Amer. Math. Soc.. 353. 5057-5069 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida and P. Polacik: "Existence of stable subharmonic solutions for reaction-diffusion equations"J. Diff. Eqs.. 169. 255-280 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] A. Yamada: "Ahlfors functions on compact bordered Riemann surfaces"J. Math. Soc. Japan. 53. 261-283 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kubota: "Stable domains of automorphisms of Banach spaces at fixed points"Bulletin of Tokyo Gakugei University. 53. (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Mizoguchi, K.Ishige: "Blow-up behavior for semilinear heat equations with Neumann boundary conditions"J.Differential Integral Equations. (to appear).

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Mizoguchi: "On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity"Math.Z. 239. 215-229 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Mizoguchi, E.Yanagida: "Life span of solutions for a semilinear parabolic problem with small diffusion"J.Math.Anal.Appl. 261. 350-368 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Mizoguchi: "Asymptotic behavior of zeros of solutions for parabolic equations"J.Differential Equations. 170. 51-67 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Mizoguchi, E.Yanagida: "Life span of solutions with large initial data in a semilinear parabolic equation"Indiana Univ.Math.J.. 50. 591-610 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] P.Polacik, E.Yanagida: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc.Cont.Dyn.Systems. 8. 209-218 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"J.Differential Equations. 179. 311-335 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Standing Pulse Solutions in Reaction-Diffusion Systems with Skew-Gradient Structure"J.Dyn.Diff.Eqs. 4. 189-205 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Stability of nonconstant stationary solutions for reaction-diffusion systems on graphs"Japan J.Indust.Appl.Math. 18. 25-42 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida, W.M.Ni, I.Takagi: "Stability of least energy patterns of the shadow system for an activator-inhibitor model"Japan J.Indust.Appl.Math.. 18. 259-272 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida, W.M.Ni, P.Polacik: "Monotonicity of stable solutions in shadow systems"Trans.Amer.Math.Soc.. 353. 5057-5069 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida, P.Polacik: "Existence of stable subharmonic solutions for reaction-diffusion equations"J.Differential Equations. 169. 255-280 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Yamada: "Ahlfors functions on compact bordered Riemann surfaces"J.Math. Soc.Japan. 53. 261-283 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Kubota: "Stable domains of automorphisms of Banach spaces at fixed points"Bulletin of Tokyo Gakugei University, Sect.IV Mathematics and Natural Sciences. 53. (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] 伊藤一郎: "中学校における情報教育(東京学芸大学附属中学校での検討課題)167-174"情報処理学会情報教育シンポジウム論文集, IPSJ Symposium Series. 9. (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Mizoguchi, E.Yanagida: "Life span of solutions with large initial data in a semilinear parabolic equation"Indiana Univ. Math. J.. 50. 591-610 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Mizoguchi, E.Yanagida: "Life span of solutions for a semilinear parabolic problem with small diffusion"J. Math. Aual. Appl.. 261. 350-368 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] N.Mizoguchi: "Asymptotic behavior of zeros of solutions for parabolic equations"J. Differential Equations. 170. 51-67 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] W.M.Ni, P.Polacik, E.Yanagida: "Monotonicity of stable solutions in shadow systems"Trans. Amer. Math. Soc.. 353. 5057-5069 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kubota: "Stable domains of automorphisms of Banach spaces at fixed points"Bull. Tokyo Gakugei Univ.. 53. 1-6 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Yamada: "Ahlfors functions on compact bordered Riemann surfaces"J. Math. Soc. Japan. 53. 261-283 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 柳田英二, 栄伸一郎: "微分方程式論"朝倉書店. 24 (2001)

    • Related Report
      2001 Annual Research Report

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

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