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Qualitative Theory of Nonlinear Elliptic Differential Equations

Research Project

Project/Area Number 09640196
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionUniversity of Tokushima

Principal Investigator

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

Co-Investigator(Kenkyū-buntansha) KOHDA Atsuhito  University of Tokushima, Faculty of Engineering, Associate Professor, 工学部, 助教授 (50116810)
NAITO Manabu  University of Ehime, Faculty of Science, Professor, 理学部, 教授 (00106791)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1998: ¥800,000 (Direct Cost: ¥800,000)
Keywordsnonlinear / elliptic / differential equations / qualitative theory / quasilinear / weak solutions / bifurcation / eigenvalue problems / 楕円形
Research Abstract

We studied the subjects related to qualitative theory of nonlinear elliptic differential equations : (i) bound-ary value problems of quasilinear elliptic equations in a bounded domain ; (ii) oscillatory problem of ordi-nary differential equations which derived from qualitative problem of elliptic equations in an unbounded domain. Our results are the following.
[1] The asymptotic behavior of eigenvalues and eigenfunctions of p-Laplace operator is investigated. We obtain the best constant of L*-Poincare inequality, and a limit equation which the limits of eigenvalues and eigenfuncitons satisfy in a weak sense.
[2] A Sturm-Liouville equation on (a, *) is examined. Supposing a strongly nonoscillatory condition, we obtain a sequence of positive princial eigenvalues and the corresponding principal eigenfunctions.
[3] Concering an initial value problem of parabolic equation, we obtain a new sufficient condition on the initial value which determines the solution to blow up. Furthermore, we investigate the asymtotic behavior of the solution at the blowing lip time.
[4] We consider a second order half-linear differential equation on [a, *]. We take up the two classes of nonoscillatory solutions (i.e., princilal solutions and non principal solutions), and show that precise information can be drawn as to how the number of zeros of these solutions changes as lambda varies from zero to infinity.
[5] A bifurcation problem or a nonlinear eigenvalue problem for degenerate quasilinear elliptic equations with Dirichelt boundary condition is studied. By virture of our estimates, we can apply the Leray-Shauder degree theory to our problem and obtain the bifurcation of nontrivial weak solutions.
[6] Quasiperiodic solutions to Van der Pol type equations driven by two or more distinct freequency input signals are considered. The existence and uniqueness results are obtained from the viewpoint of numerical analysys.

Report

(3 results)
  • 1998 Annual Research Report   Final Research Report Summary
  • 1997 Annual Research Report

Research Products

(22 results)

All Other

All Publications (22 results)

  • [Publications] N.Fukagi,M.Ito and K.Narukawa: "Limit as p→∞ of p-Laplace eigenvalue problems and L^∞-inequality of the Poincare type." Differential Integral Equations. 印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and M.Naito: "A singular eigenvalue problem for Sturm-Liouville equations." Differ.Uravn.印刷中.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Kohada and T.Suzuki: "A note on the blow-up pattern for a parabolic equation." J.Math.Tokushima Univ.32. 19-25 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Elbert,T.Kusano and M.Naito: "Singular eigenvalue problems for second order linear ordinary differential equations." Arch.Math.(Brno). 34. 59-72 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Fukagai,M.Ito and K.Narukawa: "A bifurcation problem of some nonlinear degenerate elliptic equations." Adv.Differential Equations. 2. 895-926 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Z.Ali,Y.Shinohara,他: "Existence and uniqueness of quasiperiodic solutions to Van der Pol type equations." J.Math.Tokushima Univ.31. 69-80 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Fukagai, M.Ito and K.Narukawa: "Limits as p * * of p-Laplace eigenvalue problems and related Poincare type inequalities." Differential Integral Equations. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] T.Kusano and M.Naito: "A Singular eigenvalue problem for Sturm-Liouville equations." Differ.Uravn.(in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Kohda and T.Suzuki: "A note on the blow-up pattern for a parabolic equation." J.Math.Tokushima Univ.32. 19-25 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Elbert, T.Kusano and M.Naito: "Singular eigenvalue problems for second order linear ordinary differential equations." Arch.Math. (Brno). 34. 59-72 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Fukagai, M.Ito and K.Narukawa: "A bifurcation problem of some nonlinear degenerate elliptic equations." Adv.Differential Equations. 2. 895-926 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] Z.Ali, Y.Shinohara et al.: "Existence and uniqueness of quasiperiodic solutions to Van der Pol type equations." J.Math.Tokushima Univ.31. 69-80 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Fukagai,M.Ito and K.Narukawa: "Limit as μ→∞ ofμLaplace eigenvalue problems L -inequality of the Poincare type." Differential Integral Equations. (印刷中).

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Kusano and M.naito: "A singular eigenvalue problem for Sturm-Liouville equations." Differ.Uravn.(印刷).

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Kohda and T.Suzuki: "A note on the blow-up pattern for aparabolic equation." J.Math.Tokushima Univ.32. 19-25 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Elbert,T.Kusano and M.Naito: "Singular eigenvalue problems for second order linear ordinary differential equations." Arch.Math.(Brno). 34. 59-72 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Fukagai, M.Ito, 他: "A bifurcation problem of some nonlinear degenerate elliptic equations" Adv.Diff.Equations. (印刷中).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano and M.Naito: "A singular eigenvalue problem for Sturm-Liouville equations" Differentsial'nye Uravneniya. (印刷中).

    • Related Report
      1997 Annual Research Report
  • [Publications] M.Naito: "An asymptotic theorem for a class of nonlinear neutral differential equations" Czechoslovak Math.J.(印刷中).

    • Related Report
      1997 Annual Research Report
  • [Publications] J.Kiyomura, T.Kusano, 他: "Positive solutions of second order quasilinear ordinary differential equations with general nonlinearities" Studia Sci.Math.Hungarica. (印刷中).

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Kusano, M.Naito, 他: "Second-order half-linear eigenvalue problems" Fukuoka Univ.Sci.Rep.27. 1-7 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Z.Ali, Y.Shinohara, 他: "Existence and uniqueness of quasiperiodic solutions to Van der Pol type equations" J.Math.Tokushima Univ.31. 69-80 (1997)

    • Related Report
      1997 Annual Research Report

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Published: 1998-03-31   Modified: 2016-04-21  

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