• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

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)
林 仲夫  大阪大学, 理学研究科, 教授 (30173016)
西原 健二  早稲田大学, 政治経済学部, 教授 (60141876)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥14,700,000 (Direct Cost: ¥14,700,000)
Fiscal Year 2003: ¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2002: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2001: ¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2000: ¥3,800,000 (Direct Cost: ¥3,800,000)
KeywordsNonlinear evolution equation / Nonlinear elliptic equation / Nonlinear PDE / Method of variation / Subdifferential operator / Nonlinear Evolution Equation / Nonlinear Elliptic Equation / Variational Method / 非線形放物型方程式 / 発展方程式 / 楕円型方程式 / 放物型方程式
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.

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (62 results)

All Other

All Publications (62 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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [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
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Garcia-Huidobro, R.Manasevich, M.Otani: "Existence results for p-Laplacian-like systems od O.E.D.'s"Funkcialaj Ekvacioj. 46. 253-285 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Inoue, K.Matsuura, M.Otani: "Strong solutions of magneto-micropolar fluid equation, Dynamical Systems and Differential Equations"Dynamical Systems and Differential Equations, edited by W.Fei, S.Hu \& X.Lin, American Institute of Mathematical Sciences. 439-448 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Yamada: "Multiple coexistence states for a prey-predator model with cross-diffusion"J.Differential Eauations/Elsevier. 197. 315-348 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Yamada: "Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion"Discrete and Continuous Dynamical Systems/American Institure of Mathematical Sciences. 10. 719-730 (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] L.Jeanjean, K.Tanaka: "Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities"Calculus of Variations and Partial Differential Equations. (未定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] L.Jeanjean, K.Tanaka: "A note on a mountain pass characterization of least energy solutions"Advanced Nonlinear Studies. 3. 445-455 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] H.Ishii, Paola Loreti: "Relaxation of Hamilton-Jacobi equations"Arch.Ration.Mech.Anal.. 169. 265-304 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Horie, H.Ishii: "Simultaneous effects of homogenization and vanishing viscosity in fully nonlinear elliptic equations"Funkcial.Ekvac. 46. 63-88 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Senba, T.Suzuki: "Blowup behavior of solutions to rescaled Jager-Luckhaus system"Adv.Differential Eauations. 8. 787-820 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Y.Matsuzawa, T.Suzuki, T.Tsuchiya: "Finite element approximation of H-surfaces"Math.Comp.. 72. 607-617 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Otani, Y.Sugiyama: "Lipschitz continuous solutions of some doubly nonlinear parabolic equations"Discrete and Continuous Dynamical Systems. vol.8, No.3. 647-670 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Ishiwata, M.Otani: "Concentration compactness principle at infinity with partial symmetry"Nonlinear Analysis TMA. vol.51. 391-407 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Hashimoto, M.Otani: "Sublinear elliptic equations and eigen value problems with singular coefficients"Communications in Applied Analysis. vol.6, No.4. 535-547 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Yamada et al.: "Well-posedness of one-phase Stefan problems for sublinear heat equations"Nonlinear Analysis TMA. vol.51. 587-606 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Hirose, Y.Yamada: "Multiple existence of positive solutions of competing species equations with diffusion and large interactions"Advances in Mathematical Sciences and Applications. vol.12, No.1. 435-453 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] L.Jean jean, K.Tanaka: "A positive solution for an asymptotically linear elliptic problem on R^N autonomous at infinity"ESALM Control Optim.Colc.Var.. vol.7. 597-614 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Nakashima, K.Tanaka: "Clustering layers and boundary layers in spacially inhomogeneous phase transition problems"Ann.Inst.H.Poincare Analyse Nonlineaire. vol.10. 107-143 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Hayashi, P.I.Naumkin: "Asymptotic expansion of small solutions to quadratic Schrodinger equations in two space dimensions"International Journal of Math.and Mathematical sciences. vol.29. 501-516 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Hayashi, P.I.Naumkin: "A quadratic nonlinear Schrodinger equation in one space dimension"J.Differential Equations. vol.186. 165-185 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii, T.Mikami: "A two dimensional random crystalline algorithm for Gauss currature flow"Adv.in Appl.Probab.. vol.34. 491-504 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishii, P.Loreti: "A class of stochastic optimal control problems with state constraint"Indiana Univ.Math.J.. vol.51. 1169-1198 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] R.Magnanini, S.Sakaguchi: "Matzoh ball soup : Heat conductors with a stationary isothermic surface"Annals of Math.. vol.156, No.3. 931-946 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Mitsuhara OTANI, Toshie SUGIYAMA: "A method of energy estimates in L^∞ and its applications to porous medium equations"J. Math. Soc. Japan. vol.53,No.4. 745-789 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mitsuharu OTANI, Yoshie SUGIYAMA: "C^∞-solutions of some degenerate pavabolic equations"Nonlinear Analysis TMA. vol.44. 3171-3181 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Panel DRABEK, Mitsuhara OTANI: "Global bifurcation result for the p-biharmonic operator"The Electronic Journal of Diffevential Equations. vol.2001,No.48. 1-19 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.HASHIMOTO, M.ISHIWATA, M.OTANI: "Quasilinear elliptic aquations in infinite tabe-shoped domains"Adv. Math. Sic. Appl.. (to appear).

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshio YAMADA: "Positive solutions for Latka-Valterra competition system with diffusion"Nonlinear Anelysis, TMA. vol.47. 6085-6096 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shingo TAKEUCHI, Yoshio YAMADA: "Asymptotic properties of a reaction-diffasion equation with degenerete p-Laplation"Nonliner Anelysis, TMA. vol.42. 41-61 (2000)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shinji ADACHI, Kazunaga TANAKA: "Multiple positive solutions for nonhomogeneous elliptic equations"Nonlinear Analysis, TMA. vol.47. 3783-3793 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shinji ADACHI, Kazunaga TANAKA: "Existence of positive solutions for a class of nonhomogeneous elliptic equitions in IR^N"Nonlinear Analysis, TMA. vol.48. 685-705 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeaki KOIKE, T.ISHIBSHI: "On fally nonlinear PDFs devied from vaviational problems of L^P-norms"SIAM Journal of Mathematical Aualysis. vol.33,No.3. 545-569 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeaki KOIKE, T.TAKAHASHI: "Remarks on regulanity of viscousity solutions of fulty nonlinear uniformly elliptic PDEs with measurable ingredients"Advances in Differential Equations. vol.7,No.4. 493-512 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeru SAKAGUCHI: "Stationary critical points of the heat flow in spaces of constant curvature"J. London Math. Soc.. vol.63. 400-412 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Shigeru SAKAGUCHI: "Regularity of interfaces with rign changes of solutions of one-dimensional porous medium equation"J. Differential Equations. vol.178. 1-59 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Mitsuharu OTANi: "Local existence of C^∞ solutions for porous media equations"Free Boundary Problems, T & A II, Gakuto International Series . 14. 367-375 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Michinori IsHiWATA & Mitsuharu OTANi: "Effect of critical point at infinity on the multiplictiy of solutions for singular perturbed semilinear elliptic equations in exterior domains."Free Boundary Problems, T & A I, Gakuto International Series. 13. 129-136 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Mitsuharu OTANi & Yoshie SOGIYAMA: "A method of energy estimaties in L^∞ and its application to porous medium equations"J.Math.Soc.Japan. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Mitsuharu OTANi & Yoshie SUGIYAMA: "Gradient estimates for solutions of nonlinear degenerate parabolic equations"Discrete and Continuous Dyramical Systems. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Yoshio YAMADA: "Coexistence states for Lotka-Volterra systems with cross-diffusion"Fields Institute Communications. 25. 551-564 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.ICHiKAWA & Yshio YAMADA: "Some remarks on global solutions to quasilinear parabolic system with cross-diffusion"Funkcial.Ekvac.. 43. 285-301 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kazunaga TANAKA: "Periodic solution for singular Hamiltonian systems and closed geodesics on non-compout Riemannian manifolas"Ann.Inst.H.Poincare Anal.Nonlineaire. 17. 1-33 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.AOACHi & Kazunaga TANAKA: "Trudinger type inequalition in R^N and their best exponents"Proc.Amer.Math.Soc.. 128. 2051-2057 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Kenji NiSHiHARA: "Boundary effect on stationary viseous shock wave for scalar viscous conservation laws"J.Math.Anal.Appl.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Kenji NiSHiHARA & H ZHAO: "Convergence Tate to viscous shock profile for general scalar viseous conservation lows with large initial data"J.Math.Soc.Japan. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Takashi SUZUKi: "A note on the stability of stationary solutions to a system of chemetanois"Communications in Contemporary Mathematics. 2. 373-383 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.NAGAI,T.SENBA & Takashi SUZUKi: "Chemotactic collapse in a parabolic system of mathematical biology"Hinoshima Math.J.. 30. 463-497 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shigeru SAKAGUCHi: "Regularity of the interfaces with sign changes of solutions of the 1-dimensional porous medium equitras"J.Diffenetial Equations. (in press). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Shigeru SAKAGUCHi: "Stationary critical points of the heat flow in spaces of constant curvature"J.London Math.Soc.. (in press). (2001)

    • Related Report
      2000 Annual Research Report

URL: 

Published: 2000-04-01   Modified: 2016-04-21  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi