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2003 Fiscal Year Final Research Report Summary

Study on Nonlinear Evolution Equations and Nonlinear Elliptic Equations

Research Project

Project/Area Number 12440051
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

OTANI Mitsuharu  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (30119656)

Co-Investigator(Kenkyū-buntansha) ISHII Hitoshi  Waseda University, School of Education, Professor, 教育学部, 教授 (70102887)
TANAKA Kazunaga  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
YAMADA Yoshio  Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (20111825)
SAKAGUCHI Shigeru  Ehime University, Faculty of Science, Professor, 理学部, 教授 (50215620)
SUZUKI Takashi  Osaka University, Graduate School of Engineering Science, Professor, 大学院・基礎工学研究科, 教授 (40114516)
Project Period (FY) 2000 – 2003
KeywordsNonlinear evolution equation / Nonlinear elliptic equation / Nonlinear PDE / Method of variation / Subdifferential operator
Research Abstract

(1)"L^∞-energy method" is invented. This assures the high differentiablity of solutions of quasilinear parabolic equations. By this method, the existence of W^<1. ∞>-solutions for a general doubly nonlinear parabolic equations and the open problem : "porous medium equations admit C^∞-solutions?" is solved affirmatively. Recent studies suggest that this gives a quite powerful tool for various problems.
(2)"The theory of nonmonotone perturbations for subdifferentials " is extended to Banach space setting. By this theory, we can treat the existence and regularity of solutions for degenerate parabolic equations in a more natural way than Galerkin' s method and open problems, left unsolved in the usual way, were solved.
(3)A Concentration Compactness (CC) theory with partial symmetry is given. The usual CC theory is known to be useful to analyze the problem with lack of compactness. On the other hand, the high symmetry such as the radial symmetry often recovers the compactness. It is studied how the partial symmetry not enough to recover compactness is reflected to CC theory. By this theory, the existence of nontrivial solutions is proved for some quasilinear elliptic equations in infinite cylindrical domains.
(4)The classical "Principle of Symmetric Criticality (PSC)" by R.Palais assures that under suitable conditions, critical points in the subspace with the symmetry give real critical points in the whole space, but is restricted to the system with variational structures. PSC is extended to a more general theory which covers the elliptic systems without full symmetry or evolution equations including time evolution terms.
(5)A new degree theory is established. It can teat mutivuled operators including subdifferential operators and cover nonlinear PDE with various multivaluedness nature.
(6)The theory of nonmonotone perturbations for subdifferentials is ameliorated to cover the initial-boundary value problems and time periodic problems for magneto-micropolar fluid equations.

  • Research Products

    (14 results)

All Other

All Publications (14 results)

  • [Publications] Mitsuharu Otani, Y.Sugiyama: "A method of energy estimates in L^∞ and its application to porous medium equations"J.Math.Soc.Japan. 53. 745-789 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kuto, Y.Yamada: "Multiple coexistence states for a prey-predator system with cross-diffusion"J.Differential Equations. 197. 315-348 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ding, K.Tanaka: "Multiplicity of positive solutions of a nonlinear Schrodinger equation"Manuscripta Mathematica. 112. 109-135 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] P.Loreti, H.Ishii: "Relaxation of Hamilton-Jacobi equations"Arch.Ration.Mech.Anal.. 169. 265-304 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Naito, K.Yoshida, T.Suzuki: "Self-similar solutions to a parabolic system modelling chemotaxis"J.Differential Equations. 184. 386-421 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] R.Magnanini, S.Sakaguchi: "On heat conductors with a stationary hot spot"Ann.Mat.Pura Appl.. 183. 1-23 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] N.Kenmochi, M.Otani, M.Niezgodka: "Proceedings of the second Polish-Japanese Days on : Mathematical Aspects of Modeling Structure Formation Phenomena"GAKUTO Intern.Ser.Math.Sci.Appl.Vol.17. 396 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Mitsuharu Otani, Y.Sugiyama: "A method of energy estimates in L^∞ and its application to porous medium equations"J. Math. Soc. Japan. vol.53. 745-789 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kuto, Y.Yamada: "Multiple coexistence states for a prey-predator system with cross-diffusion"J. Differential Equations. vol.197. 315-348 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Ding, K.Tanaka: "Multiplicity of positive solutions of a nonlinear Schrodinger equation"Manuscripta Mathematica. vol.112. 109-135 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Loreti, H.Ishii: "Relaxation of Hamilton-Jacobi equations"Arch. Ration. Mech. Anal.. vol.169. 265-304 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y.Naito, K.Yoshida, T.Suzuki: "Self-similar solutions to a parabolic system modeling chemotaxis"J. Differential Equations. vol.184. 386-421 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] R.Magnanini, S.Sakaguchi: "On heat conductors, with a stationary hot spot"Ann. Mat. Pura Appl.. vol.183. 1-23 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] N.Kenmochi, M.Otani, M.Niezgodka: "Proceedings of the second Polish-Japanese Days on : Mathematical Aspects of Modeling Structure Formation Phenomena"GAKUTO Intern. Ser. Math. Sci. Appl. Vol. 17.

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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