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Analysis of nonlinear diffusion equations and related phase transition problems

Research Project

Project/Area Number 12640224
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

YAMADA Yoshio  School of Science and Engineering, Professor, 理工学部, 教授 (20111825)

Co-Investigator(Kenkyū-buntansha) HIROSE Munemitsu  Meiji Univ., School of Sci. & Tech., Lecturer, 理工学部, 専任講師 (50287984)
TANAKA Kazunaga  School of Science and Engineering, Professor, 理工学部, 教授 (20188288)
OTANI Mitsuharu  School of Science and Engineering, Professor, 理工学部, 教授 (30119656)
TAKEUCHI Shingo  Gakushuin Univ., Dept. of Math., Assistant Professor, 理学部, 助手 (00333021)
NAKASHIMA Kimie  Tokyo Univ. of Fisheries, Associate Professor, 助教授 (10318800)
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: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2000: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsreaction-diffusion equation / nonlinear diffusion / global solution / Lotka-Volterra type / positive stationary solution / cross-diffusion / 反応拡散 / 大域的存在 / Shigesadaモデル / 分岐 / 解の多重性
Research Abstract

In our project we have mainly discussed the stationary and non-stationary problems for the following reaction diffusion systems with quasilinear diffusion terms:
(E) u_l = Δ[(1 + αv + γu)u] + uf (u, v), v_l = Δ[(1 + βv + δv)v] + vg (u, v).
This is a well-known system which models the habitat segregation phenomenon between two species. In (E) u, v denote the population densities and f, g represent the interaction between u and v such as Lotka-Volterra competition type or prey-predator type.
(1) Non-stationary problem. When the system has a cross-diffusion effect, the existence result of global solutions was restricted to the two dimensional case. We have proved that, if α, γ > 0 and β = δ = 0, then (E) admits a unique global solution without any restrictions on the space dimension and the amplitude of initial data. Our strategy is to decouple the system and study reaction-diffusion equations separately. We combine parabolic fundamental estimates with energy estimates of solutions of parabolic equation with self-diffusion. This method is also valid for the case δ > 0; so that the global existence is shown when the space dimension is less than six.
(2) Stationary problem. From the view-point of mathematical biology, it is very important to study positive stationary solutions and to know their number. We have tried to get some conditions for the multiplicity of such positive solutions. In particular, the multiple existence is established if interactions are very large in case of competition model with linear diffusion or if one of cross-diffusion is very large in case of prey-predator model.

Report

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

    (72 results)

All Other

All Publications (72 results)

  • [Publications] 竹内慎吾, 山田義雄: "Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian"Nonlinear Analysis, TMA. 42. 41-61 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 市川達夫, 山田義雄: "Some remarks on global solutions to quasilinear parabolic system with Cross-Diffusion"Funkcialaj Ekvacioj. 43. 285-301 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 山田 義雄: "Coexistence states for Lotka-Volterra systems with cross-diffusion"Fields Institute Communications. 25. 551-564 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 足達慎二, 田中和永: "Trudinger type inequalities in R^N and their best exponents"Proceedings of the American Mathematical Society. 128. 2051-2057 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 廣瀬宗光, 柳田英二: "Global structure of self-similar solutions in a semilinear parabolic equation"J.Math.Anal.Appl.. 244. 348-368 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 竹内慎吾: "Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation"SIAM Journal Math.Anal.. 31. 678-692 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 中島 主恵: "Stable transition layers in a balanced bistable equation"Differential Integral Equations. 12. 1025-1038 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 山田 義雄: "Positive solutions for Lotka-Volterra competition system with diffusion"Nonlinear Analysis, TMA. 47. 6085-6096 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 杉山由恵, 大谷光春: "A method of energy estimates in L^∞ and its application to porous medium equations"Journal Math.Soc.Japan. 53. 745-789 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 杉山由恵, 大谷光春: "C^∞-solutions of some degenerate parabolic equations"Nonlinear Analysis, TMA. 47. 3171-3181 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 足立慎二, 田中和永: "Multiple positive solutions for in homogeneous elliptic equations"Nonlinear Analysis, TMA. 47. 3783-3793 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 竹内 慎吾: "Positive solutions of a degenerate elliptic equation with logistic reaction"Proc.Amer.Math.Soc.. 129. 433-441 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 竹内 慎吾: "Multiplicity result for a degenerate elliptic equation with logistic reaction"J.Differential Equations. 137. 138-144 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 廣瀬健文, 山田義雄: "Multiple existence of positive solutions of competing species equations with diffusion and large interactions"Adv.Math.Sci.Appl.. 12. 435-453 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 杉山由恵, 大谷光春: "Lipschitz continuous solutions of some doubly nonlinear parabolic equations"Discrete Continuous Dynamical Systems. 8. 647-670 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 石渡通徳, 大谷光春: "Concentration compactness principle at infinity with partial symmetry and its application"Nonlinear Analysis, TMA. 51. 391-407 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 橋本哲, 大谷光春: "Sublinear Elliptic Equations and Eigenvalue Problems with Singular Coefficients"Comm.Applied Analysis. 6. 535-547 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] P.Felmer, M.Del Pino, 田中和永: "An elementary method for the construction of complex solutions for nonlinear Schrodinger equations"Nonlinearity. 15. 1653-1671 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 廣瀬宗光, 太田雅人: "Structure of Positive Radial Solutions to Scalar Field Equations with Harmonic Potential"J.Differential Equations. 178. 519-540 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.S.Choi, R.Lui, 山田義雄: "Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion"Discrete Continuous Dynamical Systems. 9. 1193-1200 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 中島主恵, 田中和永: "Clustering layers and boundary layers in spatially inhomogeneous phase transition problems"Ann.Inst.H.Poincare Anal.Non Lineaire. 20. 107-143 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.S.Choi, R.Lui, 山田義雄: "Existence of global solutions for the Shigesada-Kawasaki-teramoto model with strongly coupled cross-diffusion"Discrete Continuous Dynamical Systems. (予定). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shingo Takeuchi and Yoshio Yamada: "Asymptotic properties of a reaction-diffusion equation with degenerate Ρ-Laplacian"Nonlinear Anal. TMA. Vol. 42. 41-61 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Tatsuo Ichikawa and Yoshio Yamada: "Some remarks on global solutions to quasilinear parabolic system with cross-diffusion"Funkcial. Ekvac.. Vol. 43. 285-301 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshio Yamada: "Coexistence states for Lotka-Volterra systems with cross-diffusion"Fields Institute Communications. Vol. 25. 551-564 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shinji Adachi and Kazunaga Tanaka: "Trudinger type inequalities in R^N and their best exponents"Proc. Amer. Math. Soc.. Vol. 128. 2051-2057 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Munemitsu Hirose and Eiji Yanagida: "Global structure of self-similar solutions in a semilinear parabolic equation"J. Math. Anal. Appl.. Vol. 244. 348-368 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shingo Takeuchi: "Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation"SIAM J. Math. Anal.. Vol. 31. 678-692 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kimie Nakashima: "Stable transition layers in a balanced bistable equation"Differential Integral Equations. Vol. 12. 1025-1038 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshio Yamada: "Positive solutions for Lotka-Volterra competition system with diffusion"Nonlinear Anal. TMA. Vol. 47. 6085-6096 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshie Sugiyamada and Mitsuharu Otani: "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
      2002 Final Research Report Summary
  • [Publications] Yoshie Sugiyamada and Mitsuharu Otani: "C^∞-solutions of some degenerate parabolic equations"Nonlinear Anal. TMA. Vol. 47. 3171-3181 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shinji Adachi and Kazunaga Tanaka: "Multiple positive solutions for nonhomogeneous elliptic equations"Nonlinear Anal. TMA. Vol. 47. 3783-3793 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shingo Takeuchi: "Positive solutions of a degenerate elliptic equation with logistic reaction"Proc. Amer. Math. Soc.. Vol. 129. 433-441 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Shingo Takeuchi: "Multiplicity result for a degenerate elliptic equation with logistic reaction"J. Differential Equations. Vol. 137. 138-144 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Takefumi Hirose and Yoshio Yamada: "Multiple existence of positive solutions of competing species equations with diffusion and large interactions"Adv. Math. Sci. Appl.. Vol. 12. 435-453 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Yoshie Sugiyama and Mitsuharu Otani: "Lipshitz continuous solutions of some doubly nonlinear parabolic equations"Discrete Continuous Dynamical Systems. Vol. 8. 647-670 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Michinori Ishiwata and Mitsuharu Otani: "Concentration compactness principle at infinity with partial symmetry and its application"Nonlinear Anal. TMA. Vol. 51. 391-407 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Satoshi Hashimoto and Mitsuharu Otani: "Sublinear elliptic equations and eigenvalue problems with singular coefficients"Comm. Appl. Analysis. Vol. 6. 535-547 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] P. Felmer, M. del Pino and Kazunaga Tanaka: "An elementary method for the construction of complex solutions for nonlinear Schrodinger equations"Nonlinearity. Vol. 15. 1653-1671 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Munemitsu Hirose and Masato Ohta: "Structure of positive radial solutions to scalar field equations with harmonic potential"J. Differential Equations. Vol. 178. 519-540 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. S. Choi, R. Lui and Yoshio Yamada: "Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion"Discrete Continuous Dynamical Systems. Vol. 9. 1193-1200 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Kimie Nakashima and Kazunaga Tanaka: "Clustering layers and boundary layers in spatially inhomogeneous phase transition problems"Ann. Inst. H. Poincare Anal. Non. Lineaire. Vol. 20. 107-143 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. S. Choi, R. Lui and Y. Yamada: "Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion"Discrete 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.S.Choi, R.Lui, 山田義雄: "Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion"Discrete Continuous Dynamical Systems. (未定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] 愛木豊彦, 今井仁司, 石村直之, 山田義雄: "Well-posedness of one-phase Stefan problems for sublinear heat equations"Nonlinear Analysis. Vol.51. 587-606 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 廣瀬健文, 山田義雄: "Multiple existence of positive solutions of competing species equations with diffusion and large interactions"Adv. Math. Sci. Appl.. Vol.12. 435-453 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 廣瀬宗光, 太田雅人: "Structure of positive radial solutions to scalar field equations with harmonic potential"J. Differential Equations. Vol.178. 519-540 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 菱田俊明: "The nonstationary Stokes and Naviers-Stokes equations in aperture domains"Elliptic and Parabolic Problems(Rolduc/Gaeld 2001). 126-134 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.del Pino, P.Felmer, 田中和永: "An elementary construction of complex patterns in nonlinear Schrodinger equations"Nonlinearity. Vol.15. 1653-1671 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 山田義雄: "Positive solutions for Lotka-Volterra competition system with diffusion"Nonlinear Analysis. Vol.47. 6085-6096 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 大谷光香, 杉山由恵: "A method of energy estimates in L^∞ and its application to porous medium equations"J. Math. Soc. Japan. Vol.53, No.4. 745-789 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 足達慎二, 田中和永: "Multiple positive solutions for nonhomogeneous elliptic equations"Nonlinear Analysis. Vol.47. 3783-3793 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 足達慎二, 田中和永: "Existence of positive solutions for a class of nonhomogeneous elliptic equations in R^N"Nonlinear Analysis. Vol.48. 685-705 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 中島主恵: "Multi-layered stationary solutions for a spatially inhomogeneous Allan-Cahn equation"Nonlinear Analysis. Vol.47. 825-836 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 竹内慎吾: "Asymptotic behavior of solutions for partial differential equations with degenerate diffusion and logistic reaction"Nonlinear Analysis. Vol.47. 1715-1724 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 竹内慎吾: "Positive solutions of a degenerate elliptic equation with logistic reaction"Proc. Amer. Math. Soc.. Vol.129. 433-441 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 竹内慎吾: "Multiplicity result for a degenerate elliptic equation with logistic reaction"J. Differential Equations. Vol.137. 138-144 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 愛木豊彦, 今井仁司, 石村直之, 山田義雄: "One-phase Stefan problems for sublinear heat equations : Asymptatic behavior of solutions"Communications in Applied Analysis. (予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 愛木, 今井, 石村, 山田: "Well-posedness of one-phase Stefan problems for sublinear heat equations"Nonlinear Analysis. (予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 廣瀬健文, 山田義雄: "Multiple existence of positive solutions of competing species equations with diffusion and large interactions"Adv. Math. Sci. Appl.. (予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 中島主恵, 田中和永: "Clustering layers and boundary layers in spatially inhomogeneous phase transition problems"Ann. Inst. H. Poincare Anal. Non Lineaire. (予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 廣瀬宗光, 太田雅人: "Structure of positive radial solutions to scalar field equations with harmonic potential"J. Differential Equations. (予定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 竹内慎吾,山田義雄: "Asymptotic properties of a reaction-diffusion equation with degenerate p-Laplacian"Nonlinear Anal.Ser.A : Theory Methods. Vol.42,No.1. 41-61 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 市川達夫,山田義雄: "Some remarks on global solutions to quasilinear parabolic system with cross-diffusion"Funkcialaj Ekvacioj. Vol.43,No.2. 285-301 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 山田義雄: "Coexistence states for Lotka-Volterra systems with cross-diffusion"Fields Institule Communications. Vol.25. 551-564 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 山田義雄: "Positive solutions for Lotka-Volterra competition system with diffusion"Nonlinear Anal.Ser.A : Theory Methods. 発表予定. (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] P.Felmer,田中和永: "Scattering solutions for planar singular Hamiltonian systems via minimization"Adv.Differential Equations. Vol.5. 1519-1544 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 廣瀬宗光,柳田英二: "Global structure of self-similar solutions in a semilinear parabolic equation"J.Math.Anal.Appl.. Vol.244,No.2. 348-368 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 中島主恵: "Stable transition layers in a balanced bistable equation"Differential Integral Equations. Vol.13. 1025-1038 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 竹内慎吾: "Positive solutions of a degenerate elliptic equation with logistic reaction"Proc.Amer.Math.Soc.. Vol.129,No.2. 433-441 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 竹内慎吾: "Behavior of solutions near the flat hats of stationary solutions for a degenerate parabolic equation"SlAM J.Math.Anal.. Vol.31,No.3. 678-692 (2000)

    • Related Report
      2000 Annual Research Report

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

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