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Research on canonical forms to nonlinear elliptic boundary value problems and the global structure of all solutions including singular solutions

Research Project

Project/Area Number 12640225
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YOTSUTANI Shoji  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (60128361)

Co-Investigator(Kenkyū-buntansha) MORITA Yoshihisa  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (10192783)
NINOMIYA Hirokazu  Ryukoku Univ., Faculty of Sci. and Tech., Associate Professor, 理工学部, 助教授 (90251610)
YANAGIDA Eiji  Tohoku Univ., Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80174548)
IKEDA Tsutomu  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (50151296)
OKA Hiroe  Ryukoku Univ., Faculty of Sci. and Tech., Professor, 理工学部, 教授 (20215221)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,400,000 (Direct Cost: ¥1,400,000)
Keywordselliptic equation / radial solution / canonical form / cross-diffusion / reaction diffusion / nonlocal / 半線形楕円形方程式 / スカラー曲率方程式 / 不完全分岐 / 反応拡張方程式 / 進行波解
Research Abstract

First, a head investigator S. Yotsutani have obtained canonical forms and structure theorems for radial solutions to semilinear elliptic problems with Y. Kabeya and E. Yanagida.
Radial solutions of semilinear elliptic problems satisfy some boundary value problems for second order differential equations. It is seen that the boundary value problems can be reduced to a canonical form with the Dirichlet, Neumann or Robin boundary condition after suitable change of variables. We get structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.
Second, it is possible through the canonical forms to convert results for one problem to that of others, and moreover, to find an original methods to investigate singular solutions. As a concrete example, S. Yotsutani clarified the structure of solutions including singular solutions in a unit ball with H. Myogahara and E. Yanagida.
As related problems, we are investigating a limiting equation to a cross-diffusion equation that appears in mathematical biology. We showed that it has different kinds of singular solutions with Y. Lou and W.-M. Ni. This problem is a nonlocal nonlinear elliptic boundary problem, for which no method was known to solve it. We discovered a new method. There are a lot of interesting problems for which the method is applicable. A problem of the Ossen's spiral flow is one of them.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report

Research Products

(34 results)

All Other

All Publications

  • [Publications] H.Ikeda, K.Kondo, H.Okamoto, S.Yotsutani: "On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows"Communication on Pure and Applied Analysis. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Lou, W.-M.Ni, S.Yotsutani: "On a Limiting System in the Lotka-Volterra Competition With Cross-Diffusion"Discrete and Continuous Dynamical Systems. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Kabeya, E.Yanagida, S.Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. 1. 85-102 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Myogahara, E.Yanagida, S.Yotsutani: "Structure of positive radial solutions for semilinear Dirichlet problems on a ball"Funkcial.Ekvac.. 45. 1-21 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Kabeya, H.Norishita, H.Ninomiya: "Imperfect bifurcations arising from elliptic boundary value problems. Nonlinear Anal"Nonlinear Anal.Ser.A : Theory Methods. 48. 663-684 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Jinbo, Y.Morita: "Ginzburg-Landau Equation with magnetic effect in a thin domain"Calc.Var.Partial Differential Equations. 15. 325-352 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Ikeda, K. Kondo, H. Okamoto and S. Yotsutani: "On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows"Communication on Pure and Applied Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Lou, W.-M. Ni and S. Yotsutani: "On a Limiting System in the Lotka-Volterra Competition with Cross-Diffusion"Discrete and Continuous Dynamical Systems. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kabeya, E. Yanagida and S. Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. 1. 85-102 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Myogahara, E. Yanagida and S. Yotsutani: "Structure of positive radial solutions for semilinear Dirichlet problems on a ball"Funkcial. Ekvac.. 45. 1-21 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kabeya, H. Morishita and H. Ninomiya: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Anal. Ser. A: Theory Methods. 48. 663-684 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Jinbo and Y. Morita: "Ginzburg-Landau Equation with magnetic effect in a thin domain"Calc. Var. Partial Differential Equations. 15. 325-352 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E. Yanagida and S. Yotsutani: "A unified approach to the structure of radial solutions for semilinear elliptic problems"Japan J. Indust. Appl. Nath.. 18. 503-519 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kabeya, E. Yanagida and S. Yotsutani: "Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball"Proc. Roy. Soc. Edinburgh Sect. A. 131. 647-665 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Morishita, E. Yanagida and S. Yotsutani: "Structural change of solutions for a scalar curvature equation"Differential Integral Equations. 14. 273-288 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Yotsutani: "Canonical form related with radial solutions of semilinear elliptic equations and its applications"Taiwanese J. Math.. 5. 507-517 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Ikeda, K.Kondo, H.Okamoto, S.Yotsutani: "On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's apiral flows"Communication on Pure and Applied Analysis. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Lou, W.-M.Ni, S.Yotsutani: "On a limiting system in the Lotka-Volterra competition with cross-diffusion"Discrete and Continuous Dynamical Systems. (to appear). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Kabeya, E.Yanagida, S.Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. 1・1. 85-102 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Myogahara, E.Yanagida, S.Yotsutani: "Structure of positive radial solutions for semilinear Dirichlet problems on a ball"Funkcial.Ekvac.. 45・1. 1-21 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Kabeya, H.Morishita, H.Ninomiya: "Imperfect bifurcations arising from elliptic boundary value problems. Nonlinear Anal"Nonlinear Anal.Ser.A : Theory Methods. 48・5. 663-684 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Jimbo, Y.Morita: "Ginzburg-Landau equation with magnetic effect in a thin domain"Calc.Var.Partial Differential Equations. 15・3. 325-352 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Kabeya, E.Yanagida, S.Yotsutani: "Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems"Communication on Pure and Applied Analysis. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Lou, W.-M.Ni, S.Yotsutanl: "On a limiting system in the Lotka-Volterra competition with cross-diffusion"Differential and Continuous Dynamical Systems. (to appear). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] H.Morishita, B.Yanagida, S.Yotsutani: "Structural change of solutions for a scalar curvature equation"Differential and Integral Equations. 14・3. 273-288 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Yanagida, S.Yotsutani: "A unified approach to the structure of radial solutions for semilinear elliptic problems"Japan J. Indust. Appl. Math.. 18. 503-519 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Kabeya, B.Yanagida, S.Yotsutani: "Global structure of solutions for equations of Brezis-Nirenberg type on the unit ball"Proc.Roy. Soc. Edinburgh Sect. A. 131. 647-665 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Yotsutani: "Canonical form related with radial solutions of semilinear elliptic equations and its applications"Taiwanese J. Math.. 5. 507-517 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Yanagida and S.Yotsutani: "A unified approach to the structure of radial solutions for semilinear elliptic problems"JJIAM. 18・2(to appear). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Morishita,E.Yanagida and S.Yotsutani: "Structural change of solutions for a scalar curvature equation"Differential and Integral Equations. 14・3(to appear). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Kabeya,H.Morishita and H.Ninomiya: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Analysis TMA. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] D.Hilhorst,M.Iida,M.Mimura and H.Ninomiya: "A competition-diffusion system approximation to the classical two-phase Stefan problem"JJIAM. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Y.Morita and Y.Mimoto: "Collision and collapse of layers in a 1-D scalar reaction-diffusion equation,"Physica D. 140. 151-170 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Jimbo and Y.Morita: "Notes on the limit equation of vortex motion for the Ginzburg-Landau equation with Neumann condition"JJIAM. (to appear).

    • Related Report
      2000 Annual Research Report

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Published: 2000-03-31   Modified: 2016-04-21  

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