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Studies on singular solutions of nonlinear differential equations.

Research Project

Project/Area Number 12640196
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KUSANO Takasi  Faculty of Science, Fukuoka University Professor, 理学部, 教授 (70033868)

Co-Investigator(Kenkyū-buntansha) YAMADA Naoki  Faculty of Science, Fukuoka University Professor, 理学部, 教授 (50030789)
SAIGO Megumi  Faculty of Science, Fukuoka University Professor, 理学部, 教授 (10040403)
SUYAMA Yoshihiko  Faculty of Science, Fukuoka University Professor, 理学部, 教授 (70028223)
NAITO Manabu  Faculty of Science, Fukuoka University Professor, 理学部, 教授 (00106791)
YOSHIDA Norio  Faculty of Science, Fukuoka University Professor, 理学部, 教授 (80033934)
Project Period (FY) 2000 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsnonlinear differential equation / singular differential equation / singular solution / black hole solution / white hole solution / 特異性をもつ微分方程式
Research Abstract

As a result of our studies on two types of new singular solutions, named black hole solutions and white hole solutions, it is shown that the black hole solutions may exist for a class of second order nonlinear differential equations whose differential operators have a certain singularity, and that the white hole solutions may exist for a class of second order nonlinear differential equations whose differential operators do not admit any singularity. The result exhibits a remarkable duality existing in the nonlinear structures of the differential equations having black hole solutions and white hole solutions. Furthermore, successful attempts have been made to extend the result mentioned above to a class of two-dimensional systems of second order nonlinear differential equations, thereby characterizing the existence of singular solutions of both black hole type and white hole type for such differential systems. The problem of coexistence of black hole solutions (or white hole solutions) and other types of singular solutions (such as blowing-up solutions an extinct solutions) has also been studied.
It is an important question to have detailed information about the structure of the totality of solutions, both proper and singular, of the given nonlinear differential equations with or without singularity. Given a differential equation, the two cases occur : either it has no singular solution at all or it possesses a proper solution and a singular solution simultaneously. Half-linear equations are an example of the latter.
An in-depth investigation of qualitative properties of poper solutions of these two types of nonlinear equations has also been one of the subject in our research project.

Report

(3 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • Research Products

    (40 results)

All Other

All Publications (40 results)

  • [Publications] J.Jaros, T.Kusano: "On black hole solution of second order differential equations with a singular nonlinearity in the differential operator"Funkcialaj Ekvacioj. 43. 491-509 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kusano, T.Tanigawa: "Positive decreasing solutions of systems of second order singular differential equations"Journal of Inequalities and Applications. 5. 581-602 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kusano, M.Naito: "Singular solutions of a singular didifferential equation"Journal of Inequalities and Applications. 5. 487-496 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, T.Tanigawa: "Existence of singular solutions for a class of systems of singular differential equations"Fukuoka University Seminar Reports. 30. 167-177 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaro, T.Kusano, N.Yoshida: "Picone-type inequalities for nonlinear elliptic equations and their Applications"Journal of Inequalities and Applications. 6. 387-404 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kusano, M.Naito: "Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations"Rocky Mountain Journal of Mathematics. 31. 1039-1054 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jams, T.Kusano, N.Yoshida: "Forced superlinear oscillations via Picone's identity"Acta Mathematica Universitatis Comenianae. (to appaer).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Genaralized Picone's formula and forced oscillations in quasilinear differential equations of the second order"Archivum Mathematicum(Brno). (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Oscillation properties of solutions of a class of nonlinear parabolic equations"Journal of Computational and Applied Mathematics. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Oscillatory properties of solutions of superlinear-sublinear parabolic equations via Picone-type inequalities"Mathematical Journal of Toyama University. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Picone-type inequalities for half-linaer elliptic equations and their applications"Advances in Mathematical Science and Applications. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T.Kusano, M.Naito, F.Wu: "On the oscillation of solutions of 4-dimensional Emden-Fowler differential equations"Advances in Mathematical Science and Applications. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano: "On white hole solutions of a class of nonlinear ordinary differential equations of the second order"Funkcialaj Ekvacioj. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano, T.Tanigawa: "Existence of singular solutions for a class of two-dimensional systems of second order nonlinear differential equations"Fukuoka University Science Reports. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jams, T.Kusano, T.Tanigawa: "Nonoscillation theory of second order half-linear differential equations in the framework of regular variation"(Submitted for publication).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano: "On black hole solution of second order differential equations with a singular nonlinearity in the differential operator"Funkcialaj Ekvacioj. 43. 491-509 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Kusano, T. Tanigawa: "Positive decreasing solutions of systems of second order singular differential equations"Journal of Inequalities and Applications. 5. 581-602 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Kusano, M. Naito: "Singular solutions of a singular didifferential equation"Journal of Inequalities and Applications. 5. 487-496 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, T. Tanigawa: "Existence of singular solutions for a class of systems of singular differential equations"Fukuoka University Seminar Reports. 30. 167-177 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, N. Yoshida: "Picone-type inequalities for nonlinear elliptic equations and their Applications"Journal of Inequalities and Applications. 6. 387-404 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Kusano, M. Naito: "Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations"Rocky Mountain Journal of Mathematics. 31. 1039-1054 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, N. Yoshida: "Forced superlinear oscillations via Picone's identity"Acta Mathematica Universitatis Comenianae. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, N. Yoshida: "Generalized Picone's formula and forced oscillations in quasilinear differential equations of the second order"Archivum Mathematicum(Brno). (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, N. Yoshida: "Oscillation properties of solutions of a dass of nonlinear parabolic equations"Journal of Computational and Applied Mathematics. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, N. Yoshida: "Oscillatory properties of solutions of superlinear-sublinear parabolic equations via Piconetype inequalities"Mathematical Journal of Toyama University. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, N. Yoshida: "Picone-1ype inequalities for half-linaer elliptic equations and their applications"Advances in Mathematical Science and Applications. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] T. Kusano, M. Naito, F. Wu: "On the oscillation of solutions of 4-dimensional E mden-Fowler differential equations"Advances in Mathematical Science and Applications. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaro, T. Kusano: "On white hole solutions of a class of nonlinear ordinary differential equations of the second order"Funkcialaj Ekvacioj. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, T. Tanigawa: "Existence of singular solutions for a class of two-dimensional systems of second order nonlinear differential equations"Fukuoka University science Reports. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano: "Remarks on the existence of regularlu varying solutions for second order linear differential equations"subunitted for pubbication.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J. Jaros, T. Kusano, T. Tanigawa: "Nonoscillation theory of second order half-linear differential equations in the framework of regular variation"subunitted for pubbication.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] J.Jaros, T.Kusano: "On white hole solutions of a class of nonlinear differential equations of the second order"Funkcialaj Ekvacioj. (近刊).

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Jaros, T.Kusano, T.Tanigawa: "Existence of singular solutions for a class of two-dimensional systems of second order nonlinear differential equations"Fukuoka University Science Reports. (近刊).

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Picone-type inequalities for half-linear elliptic equations and their applications"Advances in Mathematical Sciences and Applications. (近刊).

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Genaralized Picone's formula and forced oscillations in quasilinear differential equations of the second order"Archivum Mathematicum(Brno). (近刊).

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Oscillation properties of solutions of a class of nonlinear parabolic equations"Journal of Computational and Applied Mathematics. (近刊).

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Jaros, T.Kusano, N.Yoshida: "Oscillatory properties of solutions of superlinear-sublinear parabolic equations via Picone-type inequalities"Mathematical Journal of Toyama University. (近刊).

    • Related Report
      2001 Annual Research Report
  • [Publications] J.Jaros,T.Kusano and T.Tanigawa: "Existence of singular solutions for a class of systems of singular differential equations"Fukuoka Univ.Sci.Rep.. 30. 169-177 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] I.K.Kiguradze and T.kusano: "On the existence and uniqueness of periodic solutions for nonautonomous differential equations"Differentsial'nye Uravneniya. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] I.K.Kiguradze and T.Kusano: "On periodic solutions of even order ordinary differential equations"Ann.Math.Pura.Appl.. (to appear).

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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