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THE MATHEMATICAL ANALYSIS TO NON-LINEAR PHENOMENA THROUGH NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS

Research Project

Project/Area Number 09640276
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionThe University of Tokushima

Principal Investigator

ITO Masayuki  The Univ.of Tokushima, Dept.of Math.& Natural Sc., Associate P., 総合科学部, 助教授 (70136034)

Co-Investigator(Kenkyū-buntansha) KODA Atsuhito  The Univ.of Tokushima, Fac.of Technology, Associate p., 工学部, 助教授 (50116810)
MURAKAMI Koichi  The Univ.of Tokushima, Dept.of Math.& Natural Sc., Lecturer, 総合科学部, 講師 (90219890)
FUKAGAI Yoshinobu  The Univ.of Tokushima, Fac.of Technology, Assosiaite P., 工学部, 助教授 (90175563)
NARUKAWA Kimiaki  Naruto Edu.Univ., Fac.of School Ed., Professor, 学校教育学部, 教授 (60116639)
YAMADA Yoshio  Waseda Univ.Dept.of Mathematics, Professor, 理工学部, 教授 (20111825)
小野 公輔  徳島大学, 総合科学部, 講師 (00263806)
Project Period (FY) 1997 – 1998
Project Status Completed (Fiscal Year 1998)
Budget Amount *help
¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 1998: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 1997: ¥2,100,000 (Direct Cost: ¥2,100,000)
Keywordsp-Laplacian / *-Laplacian / limit eigenvalue problem / Poincare inequality / degenerate elliptic equation / reaction-diffusion equations / delay differential equation / blow up / 非線形偏微分方程式 / 退化楕円型作用素 / p-Laplacian / 非線形拡散 / キルヒホッフ / 非線形波動方程式 / 非線形振動
Research Abstract

1) The p-Laplace operators is well known as the non-linear modification of the usual Laplacian. These operators or their perturbed operators arise in the model equations for the elastic membrane, nonlinear diffusion phenomena and so on. Moreover, the limit state of solutions at p infinity is of great interest from the mathematical or technological view points. The eigenvalue problem of p-Laplacian has been studied by many authors. Since this problem can be dealt with as a variational problem, many results has been known. However, its limit problem at p infinity had been known because it cannot be described in a variatinal problem. We formulate such problem using the notion of the viscosity solution and obtain some results for the limit eigenvalues and the associate eigenfunctions.
2) In the ecological model, a reaction-diffusion equation has the nonlinear diffusion with the p-Laplace operator when the diffusion depends on the population pressure nonlinearly. Yamada has studied such equation and obtain the unique and global existence of a solution and sonic results on the set of stationary solutions. He also study the 3 species cooperative competition-diffusion systems with linear diffusion, and obtain the necessary and sufficient condition to the existence of the coexistence solutions.
3) Murakami has studied the asymptotic behavior of the solution for several higher dimensional delay differential equations and obtain the existence of periodic solutions which are bifurcated from the equilibrium, in particular, the explicit expressions of the bifurcated periodic solutions.
4) Kohda has obtained some conditions on initial value for parabolic problem which guarantee the blow-up of a solution. Moreover, he had shown the behavior of blow-up solution near blow-up time, that is blow-up patterns.

Report

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

Research Products

(21 results)

All Other

All Publications (21 results)

  • [Publications] N.Fukagai,M.Ito & K.Narukawa: "Limit as p→∞ of p-Laplace eigenvalue problems and L^∞-inequality of the Poincere type" Diff.Int.Equations. (in press).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] S.Takeuchi & Y.Yamada: "Asymptotic properties of reaction-diffusion equation with degenerate p-Laplacian" Nonlinear Analysis,Theory,Methods & Applications. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Yoshida & Y.Yamada: "Global attractivity of coexistence states for a certain class of reaction diffusion systems with 3×3 …" Advances in Mathematical Sciences and Applications. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Murakami: "Scable Periodic Solutions for Two-dimensional Linear Delay Differential Equations" Journal of Mathematical Analysis and Applications. 205・2. 512-530 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Murakami: "Asymptotic Constancy for Systems of Delay Differential Equations" Nonlinear Analysis,T.M.A.30・7. 19-25 (1997)

    • Description
      「研究成果報告書概要(和文)」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(和文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] N.Fukagai, M.Ito & K.Narukawa: "Limit as p * * of p-Laplace eigenvalue problems and L^*-inequality of the Poincare type" Diff.Int.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] S.Takeuchi & Y.Yamada: "Asymptotic properties of a reaction - diffusion equation with degenerate p - Laplacian" Nonlinear Analysis, Theory, Methods & Applications.(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] A.Yoshida & Y.Yamada: "Global attractivity of coexistence states for a certain class of reaction diffusion syatems with 3 * 3 cooperative matrices" Advances in Mathematical Sciences and Applications. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Murakami: "Stable Periodic Solutions for Two-dimensional Linear Delay Differential Equations" Journal of Mathematical Analysis and Applications. 205.2. 512-530 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1998 Final Research Report Summary
  • [Publications] K.Murakami: "Asymptotic Constancy for Systems of Delay Differential Equations" Nonlinear Analysis, T.M.A.30.7. 4595-4606 (1997)

    • 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] N.Fukagai,M.Ito&K.Narukawa: "Limit asρ→∞ of p-Laplace elgenvalue problems and -inequality of the Poincare type" Diff.Int.Equations. (in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] S.Takeuti&Y.Yamada: "Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian" nonlinear Analysis,Theory,Methods&Applications. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Yoshida&Y.Yamada: "Global attractivity of coexistence atates for a certain class of reaction diffusion systems with 3×3・・・" Advances in Mathematical Sciences and Applications. (to appear).

    • Related Report
      1998 Annual Research Report
  • [Publications] A.kobda and T.suzuki: "Anote on the blow-up pattern for a parabolic equation" J.Math.Tokushima Univ.32. 19-25 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Fukugai, M.Ito & K.Narukawa: "A Bifurcation Problem of Some Nonlinear Digenerate Elliptic Equations" Adv.Diff.Equatios. (in press).

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Yamada: "Coexistence State for Some Populatin Models with Nonlinear Cross-diffusion" Forma. 12,2. 153-166 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Ono: "Global existence,decay,and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings" Journal of Differential Equations. 137,2. 273-301 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Ono: "On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation" Journal of Mathematical Analysis and Applications. 216,1. 321-342 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Z.Ali, Y.Shinohara, H.Imai, A.Kohda, et al.: "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: 1997-03-31   Modified: 2016-04-21  

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