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2005 Fiscal Year Final Research Report Summary

Investigation of the global bifurcations/imperfect bifurcations in nonlinear elliptic partial differential equations

Research Project

Project/Area Number 15540211
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionOsaka Prefecture University (2005)
University of Miyazaki (2003-2004)

Principal Investigator

KABEYA Yoshitsugu  Osaka Prefecture University, Department of Mathematical Sciences, Associate professor, 工学研究科, 助教授 (70252757)

Co-Investigator(Kenkyū-buntansha) TSUJIKAWA Tohru  University of Miyazaki, Department of Applied Physics, Professor, 工学部, 教授 (10258288)
SENBA Takasi  University of Miyazaki, Department of Applied Physics, Professor, 工学部, 教授 (30196985)
YAZAKI Shigetoshi  University of Miyazaki, Department of Applied Physics, Associate Professor, 工学部, 助教授 (00323874)
Project Period (FY) 2003 – 2005
KeywordsElliptic partial differential equations / Bifurcation diagram / imperfect bifurcation / radially symmetric solution / multiplicity of solutions
Research Abstract

From 2003 to 2005, we investigated on the global bifurcation diagrams on the nonlinear elliptic partial differential equations. Especially, we are concerned with the imperfect bifurcations arising in the Brezis-Nirenberg equation or scalar-field equations with the Robin condition. Due to the Robin condition, we observed several specific properties of the bifurcation diagrams. The diagram has some bending point, which implies the existence of multiple solutions to the equation which we consider. Moreover, we investigated how the bifurcation branch goes to infinity in the function space and we also show the difference between the three dimensional case and the higher dimensional cases. In the three dimensional case, the bending point and the blow-up point move to left in the parameter axis continuously as the parameter in the Robin condition increases. However, if the dimension is not Less than four, then the blow-up point is the origin in the parameter space. We also showed the existence of a threshold value for which the bending occurs. For higher dimensional case, the publishing of the results is not in time, however, the journal acknowledges that our paper will appear in summer 2006.
For other co-investigators, they helped me via private communications and encouraged me to research this field although there is no joint work with them.

  • Research Products

    (23 results)

All 2006 2005 2004 2003 Other

All Journal Article (23 results)

  • [Journal Article] Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition2006

    • Author(s)
      Yoshitsugu Kabeya
    • Journal Title

      Discrete and Continuous Dynamical Systems 14・1

      Pages: 117-134

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition2006

    • Author(s)
      Yoshitsugu kabeya
    • Journal Title

      Discrete and Continous Dynamical Systems 14-1

      Pages: 117-134

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Multiplicity of solutions to a scalar-field equation involving the Sobolev critical exponent with the Robin condition2005

    • Author(s)
      Yoshitsugu Kabeya
    • Journal Title

      Advances in Differential Equations 10・10

      Pages: 1111-1129

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Numerical computations and pattern formation for adsorbate-induced phased transition model2005

    • Author(s)
      Yasuhiro Takei, Tohru Tsujikawa, Atsushi Yagi
    • Journal Title

      Scientiae Mathematicae Japonicae 61・3

      Pages: 55-64

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Chemotaxis and growth system with singular sensitivity function2005

    • Author(s)
      Masashi Aida, Koichi Osaki, Tohru Tsujikawa, Atsushi Yagi, Masayasu Mimura
    • Journal Title

      Nonlinear Analysis : Real World Applications 6

      Pages: 323-336

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains2005

    • Author(s)
      Takasi Senba
    • Journal Title

      Funkcialaj Ekvacioj 48・2

      Pages: 247-271

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Multiplicity of solutions to a scalar-field equation involving the Sobolev critical exponent with the Robin condition2005

    • Author(s)
      Yoshitsugu Kabeya
    • Journal Title

      Advances in Differential Equations 10-10

      Pages: 1111-1127

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Numerical computations and pattern formation for adsorbate-induced phased transition model2005

    • Author(s)
      Yasuhiro Takei, Tohru Tsujikawa, Atsushi Yagi
    • Journal Title

      Scientiae Mathematicae Japonicae 61-3

      Pages: 55-64

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains2005

    • Author(s)
      Takasi Senba
    • Journal Title

      Funkcialaj Ekvacioj 48-2

      Pages: 247-271

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Uniqueness and profile of solutions to a superlinear elliptic equation2004

    • Author(s)
      Yoshitsugu Kabeya, Eiji Yanagida
    • Journal Title

      Advances in Differentail Equations 9・7&8

      Pages: 771-796

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Global solution to a reaction diffusion phase transition System in R^22004

    • Author(s)
      Koichi Osaki, Yasuhiro Takei, Tohru Tsujikawa
    • Journal Title

      Advances in Mathematical Sciences and Applications 14・2

      Pages: 559-576

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Chemotactic collapse of radial solutions to Jager-Luckhaus system2004

    • Author(s)
      Takasi Senba, Takashi Suzuki
    • Journal Title

      Advances in Mathematical Sciences and Applications 14・1

      Pages: 241-250

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Two examples of noncovex self-similar solution curves for a crystalline curvature flow2004

    • Author(s)
      Tetsuya Ishiwata, Takeo K.Ushijima, Hiroki Yagisita, Shigetoshi Yazaki
    • Journal Title

      Proceedings of Japan Academy Series A 80・8

      Pages: 151-154

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Convergence of a crystalline approximation for an area-preserving motion2004

    • Author(s)
      Takeo K.Ushijima, Shigetoshi Yazaki
    • Journal Title

      Journal of Computational and Applied Mathematics 166

      Pages: 427-452

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Uniqueness and profile of solutions to a superlinear elliptic equation2004

    • Author(s)
      Yoshitsugu Kabeya, Eiji Yanagida
    • Journal Title

      Advances in Differential Equations 9-7&8

      Pages: 771-796

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Global solution to a reaction diffusion phase transition System in R^22004

    • Author(s)
      Koichi Osaki, Yasuhiro Takei, Tohru Tsujikawa
    • Journal Title

      Advances in Mathematical Sciences and Applications 14-2

      Pages: 559-576

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Chemotactic collapse of radial solutions to Jager-Luckhaus system2004

    • Author(s)
      Takasi Senba, Takashi Suzuki
    • Journal Title

      Advances in Mathematical Sciences and Applications 14-1

      Pages: 241-250

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Two examples of nonconvex self-similar solution curves for a crystalline curvature flow2004

    • Author(s)
      Tetsuya Ishiwata, Takeo K.Ushijima, Hiroki Yagisita, Shigetoshi Yazaki
    • Journal Title

      Proceedings of Japan Academy Series A 80-8

      Pages: 151-154

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Blowup behavior of solutions to the rescaled Jager-Luckhaus System2003

    • Author(s)
      Takasi Senba, Takashi Suzuki
    • Journal Title

      Advances in Differential Equations 8・7

      Pages: 787-820

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion2003

    • Author(s)
      Tetsuya Ishiwata, Shigetoshi Yazaki
    • Journal Title

      Journal of Computational and Applied Mathematics 159

      Pages: 55-64

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Blowup behavior of solutions to the rescaled Jager-Luckhaus System2003

    • Author(s)
      Takasi Senba, Takashi Suzuki
    • Journal Title

      Advances in Differential Equations 8-7

      Pages: 787-820

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Uniqueness of positive solutions to a higher dimensional Brezis-Nirenberg equation with the Robin condition

    • Author(s)
      Yoshitsugu Kabeya
    • Journal Title

      Advances in Mathematical Analysis and Applications (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Uniqueness of positive solutions to a higher dimensional Brezis-Nirenberg equation with the Robin condition

    • Author(s)
      Yoshitsugu Kabeya
    • Journal Title

      Advances in Mathematical Analysis and Applications (To appear)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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