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Qualitative Theory of Quasilinear Degenerate Elliptic Differential Equations

Research Project

Project/Area Number 11640206
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTokushima University

Principal Investigator

FUKAGAI Nobuyoshi  Tokushima University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (90175563)

Co-Investigator(Kenkyū-buntansha) ITO Masayuki  Tokushima University, Department of Mathematics and Computer Sciences, Professor, 総合科学部, 教授 (70136034)
NARUKAWA Kimiaki  Naruto University of Education, Department of Mathematics, Professor, 教授 (60116639)
NAITO Manabu  Ehime University, Faculty of Science, Professor, 理学部, 教授 (00106791)
KOHDA Atsuhito  Tokushima University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50116810)
Project Period (FY) 1999 – 2000
Project Status Completed (Fiscal Year 2000)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 1999: ¥2,500,000 (Direct Cost: ¥2,500,000)
Keywordsnonlinear / elliptic / differential equations / qualitative theory / quasilinear / weak solutions / bifurcation / eigenvalue problems / 非線形
Research Abstract

We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) boundary value problems of quasilinear elliptic equations in a bounded domain ; (ii) blowup phenomena of seminilear parabolic equations ; (iii) solutions of ordinary differential equations derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.
1. Positive solutions of a class of nonlinear eigenvalue problems are investigated. For a quasilinear elliptic problem (^*)-div (φ(|∇u|) ∇u)=λf(x, u) in Ω, u=0 on ∂Ω, we assume asymptotic conditions on φ and f such as φ(t)〜t^<p_0-2>, f(x, t)〜t^<q_0-1> as t→0 andφ(t)〜t^<p_1-2>, f(x, t)〜t^<q_1-1> as t→∞. The combined effects of sub-nonlinearity (p_0>q_0) and super-nonlinearity (p_1<q_1) imply the existence of at least two positive solutions of (^*) for 0<λ<Λ.
2. Concerning an initial value problem of semilinear parabolic equations with convex nonlinearities, we obtain sufficient conditions which determine the solution to blow up. The criteria are formulated in terms of super-solution and sub-solution of the related stational problem.
3. A Sturm-Liouville equation on [α, ∞) is examined. (1) Higher order ordinary differential equations with general nonlinearities and determination of the Kiguraze class of positive sotuions. (2) Oscillation problem of first order 4-dimensional system of ordinary differential equations. (3) Eigenvalue problem of second order half-linear ordinary differential equations on [α, ∞).

Report

(3 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report

Research Products

(14 results)

All Other

All Publications (14 results)

  • [Publications] A.Kohda: "Blow-up criteria for semiliniear parabolic equations"J.Math.Anal.Appl.. 243. 127-139 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] M.Naito: "Positive solutions of higher order ordinary differential equations with general nonlinearities"J.Math.Anal.Appl.. 250. 27-48 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kusano: "On the oscillation of solutions of 4-dimensional Emden-Fowler differential systems"Adv.Math.Sci.Appl.. 11(印刷中). (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Kusano: "Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations"Rocky Mountain J.Math.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Atsuhito Kohda and Takashi Suzuki: "Blow-up criteria for semiliniear parabolic equations."J.Math.Anal.Appl.. 243. 127-139 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Manabu Naito and Kimihiko Yano: "Positive solutions of higher order ordinary differential equations with general nonlinearities."J.Math.Anal.Appl.. 250. 27-48 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kusano Takasi, Manabu Naito and Wu Fentao: "On the oscillation of solutions of 4-dimensional Emden-Fowler differential systems."Adv.Math.Sci.and Appl.. 11 (in press). (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] Kusano Takasi and Manabu Naito: "Sturm-Liouville eigen-value problems for half-linear ordinary differential equations."Rocky Mountain J.Math.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] A.Kohda: "Blow-up criteria for semiliniear parabolic equations"J.Math.Anal.Appl.. 243. 127-139 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Naito: "Positive solutions of higher order ordinary differential equations with general nonlinearities"J.Math.Anal.Appl.. 250. 27-48 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kusano: "On the oscillation of solutions of 4-dimensional Emden-Fowler differential systems"Adv.Math.Sci.Appl.. 11(印刷中). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Kusano: "Sturm-Liouville eigenvalue problems for half-linear ordinary differential equations"Rocky Mountain J.Math.. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] A.Elbert: "On the number of zeros of nonoscillatory solutions to second-order half-linear differential equations"Ann.Univ.Sci.Budapest.Etotvos Sect.Math.. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Kusano: "Singular solutions of a singular differential equation"Journal of Inequalities and Applications. (掲載予定).

    • Related Report
      1999 Annual Research Report

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Published: 1999-03-31   Modified: 2016-04-21  

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