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Development of reaction-diffusion systems - Studies of singular limit methods -

Research Project

Project/Area Number 12304006
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionHIROSHIMA UNIVERSITY

Principal Investigator

MIMURA Masayasu  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50068128)

Co-Investigator(Kenkyū-buntansha) YANAGIDA Eiji  Tohoku University Graduate School of Sciences, Professor, 大学院・理学研究科, 教授 (80174548)
SAKAMOTO Kunimochi  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (40243547)
OHTA Takao  Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50127990)
IKEDA Hideo  Toyama University, Faculty of Sciences, Professor, 理学部, 教授 (60115128)
MORITA Yoshihisa  Ryukoku Univ., Faculty of Science and Technolgy, Professor, 理工学部, 教授 (10192783)
辻川 亨  宮崎大学, 工学部, 教授 (10258288)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥42,000,000 (Direct Cost: ¥35,400,000、Indirect Cost: ¥6,600,000)
Fiscal Year 2002: ¥15,210,000 (Direct Cost: ¥11,700,000、Indirect Cost: ¥3,510,000)
Fiscal Year 2001: ¥13,390,000 (Direct Cost: ¥10,300,000、Indirect Cost: ¥3,090,000)
Fiscal Year 2000: ¥13,400,000 (Direct Cost: ¥13,400,000)
KeywordsSingular limit methods / Interfacial dynamics / Singular perturbation method / Blow up problem / Traveling wave solutions / Equations of curvature / Chemotaxis equations / Center manifold theory / 反応拡散系の数値計算 / 進行波 / 界面運動の数値解析 / ホモクリニック力学系 / 進行波の反射現象 / 曲率方程式解析 / 走化性モデル解析 / スパイク解の運動 / 自由境界問題 / 不安定成長系
Research Abstract

Reaction-diffusion systems have been encountered in many fields of natural sciences. In this project, we develop singular limit procedures and comtempolarily numerical methods in order to investigate dynamics of patterns, forms, interfaces and singularities in such systems. In the below, we show our results obtained in this project:
(1) Derivation of a free boundary problem from competition-diffusion systems by using singular limit procedures
(2) Qualitative study of spiky solutions in biological systems by singular limit procedures.
(3) Stability analysis of skew-gradient reaction-diffusion systems.
(4) Interaction of traveling pulses and spots by using dynamical system theory
(5) Traveling wave solutions of reaction diffusion systems by using singular perturbation methods
(6) analysis of motion of mean curvatures in higher dimensional spaces.
(7) Analysis of aggregating patterns in diffusion-chemotaxis systems
(8) Numerical simulations of complex spatio-temporal patterns in reaction-diffusion systems
(9) Development of adoptive FEM methods to reaction-diffusion systems

Report

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

    (55 results)

All Other

All Publications (55 results)

  • [Publications] M.Mimura: "Annihilation dynamics in the KPP-Fisher equations"European J.Appl.Math.. Vol.13. 195-204 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Mimura: "Pulse-pulse interaction in reaction-diffusion systems"Physica D. 165. 176-198 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Ohta: "Self-replication of a pulse in excitable reaction-diffusion systems"Phys.Rev.. E66. 036218 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Ohta: "Time-modulated oscillatory structures in phase-separating reactive mixtures"Phys.Rev. E. (in press). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Morita: "Ginzburg-Landau Equation with Magnetic Effect in a Thin Domain"Calc.Var.Partial Differential Equations. Vol.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] K.Sakamoto: "Bifurcation of transition layers in reaction-diffusion systems"数理解析研究所講究録. 1249. 90-102 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida: "Standing pulse solutions in reaction-diffusion systems with skew gradient structure"J.Dyn.Diff.Eqs.. 4. 189-205 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] E.Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"J.Diff.Eqs.. 179. 311-335 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.-I.Ei: "The motion of weakly interacting pulses in reaction-diffusion systems"J.Dyn.Diff.Eqs.. 14(1). 85-137 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.-I.Ei: "Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions"Japan J.Ind.Appl.Math.. 19(2). 181-226 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Tsujikawa: "Exponential attractor for an absorbate-induced phase transition model"Kyusyu J.Math.. 56. 313-336 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Ishii: "A class of stochastic optimal control problems with state constraint"Indiana Univ.Math.J.. 51. 1169-1198 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Mimura: "Annihilation dynamics in the KPP-Fisher equations"European J. Appl, Math.. 13. 195-204 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Mimura: "Pulse-pulse interaction in reaction-diffusion systems"Physica D. 165. 176-198 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Ohta: "Self-replication of a pulse in excitable reaction-diffusion systems"Phys. Rev.. E66. 036218 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Ohta: "Time-modulated oscillatory structures in phase-separating reactive mixtures"Phys. Rev. E. In press. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Morita: "Ginzurg-Landau Equation with Msignetic Effect in a Thin Domain"Calc. Van Partial Diff. Wqns. 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] K. Sakamoto: "Bifurcation of transition layers in Reaction-diffusion systems"RIMS Kokyuroku, Kyoto Univ.. 1249. 90-102 (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: "Standing pulse solutions in reaction-diffusion systems with skew gradient structure"J. Dyn. Diff. Eqs.. 4. 189-205 (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: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"J. Diff. Eqs.. 179. 311-335 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.-I. Ei: "The motion of weakly interacting pulses in reaction-diffusion systems"J.Dyn. Diff. Eqs.. 14(1). 85-137 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.-I. Ei: "Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions"Japan J. Ind. Appl. Math.. 19(2). 181-226 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Tsujikawa: "Exponential attractor for an absorbate-induced phase transition model"Kyushu J. Math.. 56. 313-336 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Ishii: "A class of stochastic optimal control problems with state constraint"Indian Univ. Math. J.. 51. 1169-1198 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Mimura: "Annihilation dynamics in the KPP-Fisher equation"European J. Appi. Math.. Vol.13. 195-204 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Mimura: "Vanishing latent heat limit in a Stefan-like problem arising in Biology"Nonlinear Analysis, Series A. (In press). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Ohta: "Target patterns in phase separation induced by cross-linking"Journal of Physical Society of Japan. 70. 729-732 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Ohta: "Time-modulated oscillatory structures in phase-separating reactive mixtures"Phys. Rev.E. (In press).

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Morita: "Ginzburg-Landau Equation with Magnetic Effect in a Thin Domain"Calc. Var. Partial Differential Equations. Vol.15. 325-352 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Sakamoto: "Bifurcation of transition layers in reaction-diffusion systems"数理解析研究所講究録. 1249. 90-102 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Stable subharmonic solutions of reaction-diffusion equations on an arbitrary domain"Disc. Cont. Dyn. Systems. Vol.8. 209-218 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] E.Yanagida: "Mini-maximizers in reaction-diffusion systems with skew-gradient structure"Diff. Eqs.. 179. 311-335 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.-l.Ei: "The motion of weakly interacting pulses in reaction-diffusion systems"J.D.D.E.. 14(1). 85-137 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.-l.Ei: "Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimension"Japan J. Ind. AppI. Math. 19(2). 181-226 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Tsujikawa: "Exponential attractor for an absorbate-induced phase transition model"Kyusyu J. Math. Vol.56. 313-336 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Ishli: "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] M.Mimura: "A competition-diffusion system approximation to the classical two-phase Stefan problem"Japan J. Industrial and Appl. Mathematics. Vol. 18. 161-180 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Mimura: "Pulse-pulse interaction in reaction-diffusion systems"Physica D. (in press). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Ohta: "Pulse dynamics in a reaction-diffusion system"Physica D. 151. 61-72 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Ohta: "Self-Propulsion of Cellular Structures in Chemically Reacting Mixtures"Physical Review. E64. 045201-1-045201-4 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Yanagida: "Life span of solutions with large initial data in a semilinear parabolic equation"Indiana Math. J.. 50. 591-610 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Yanagida: "Monotonicity of stable solutions in shadow systems"Trans. Amer. Math. Soc.. 353. 5057-5069 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Morita: "Breaking Homoclinic Bifurcations in Diffusively Coupled Equations"J. Dynamics and Differential Equations. Vol. 13. 613-649 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Tsujikawa: "Exponential attractor for a chemotaxis-growth system of equations"Nonlinear Analysis. (in press).

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Sakamoto: "Internal Layers Intersecting the Boundary of Domain in the Allen-Cahn Equation"Japan Journal of Industrial and Applied Mathematics. Vol. 18・No. 3. 697-738 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 柳田英二, 榮伸一朗: "微分方程式論"朝倉書店. 1-224 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] M.Mimura: "Hormone-dependent model on seed germination sensitive to growth stage"J.Phys.Jpn.. 69-4. 460-470 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Mimura: "Reaction-diffusion modelling of bacterial colony pattems"Physica A. 282. 283-303 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Mimura: "Dynamics of travelling breathers arising in reaction-diffusion systems-ODE modelling approach-"Hiroshima Math.J.. 30. 221-256 (2000)

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

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Ikeda: "Bifurcation phenomena from standing pulse solutions of bistable reaction-diffusion systems"Journal of Dynamics and Differential Equations. Vol.12. 117-167 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.-I.Ei.: "Renormalization-group method for reduction of evolution equations ; invariant manifolds and envelopes"Annals of Physics. 280. 236-298 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] E.Yanagida: "Extinction and blowup of positive radial solutions for a semilinear elliptic equation,"Nonlinear Anal.. 39. 365-377 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] E.Yanagida: "Existence of stable subharmonic solutions for reaction-diffusion equations."J.Differential Equations. 169. 255-280 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] 三村昌泰: "生物の形づくりの数理と物理(第3章執筆)"共立出版株式会社. 1-48 (2000)

    • Related Report
      2000 Annual Research Report

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

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