Advertisement

Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit

  • Yasumasa Nishiura
Part of the Dynamics Reported book series (DYNAMICS, volume 3)

Abstract

Recognition stems from realization of the separation boundary between two different physical or chemicalstates. In other words we can observe natural phenomena through the emergence and evolution of theinterface between these states as in solidification, combustion, chemical reaction, and biological patterns. The interface studied here results from the balance between two opposing tendencies: adiffusive effect and a (physical or chemical)separation kinetics built in the system. The former attempts to smooth out the inhomogeneity as in the heat equation, and the latter drives the system to one or the other pure state such as solid or liquid (see, for instance, Fife [19] for details). Turing’s contribution [58] is one of the pioneering works related to the onset of spatial patterns through a cooperative work of diffusion and separation kinetics. Besides the existence of these two tendencies, another key ingredient to produce interesting interfacial patterns is the differences of the strength of the above mixing and unmixing effects among species involved in the system. In fact, reaction diffusion systems for two componentsu and v, which are the main concern in this paper, can be classified formally as
  1. (i)

    There is a difference in the diffusion rates of u and v;

     
  2. (ii)

    There is a difference in the reaction rates of u and v;

     
  3. (ii)

    There are differences in the diffusion and reaction rates of u and v, i.e., a combination of (i) and (ii).

     

Keywords

Stable Solution Reaction Diffusion System Principal Eigenvalue Singular Limit Slow Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Alexander, R. Gardner and C. Jones, A topological invariant arising in the stability analysis of travelling waves, J. reine angew. Math.,410 (1990), 167–212.MathSciNetMATHGoogle Scholar
  2. [2]
    N. D. Alikakos, P. W. Bates, and G. Fusco, Slow motion for the Cahn- Hilliard equation in one space dimension, J. Differential Equations,90 (1991), 81–135.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    L. Bronsard and R. V. Kohn, On the slowness of the phase boundary motion in one space dimension, Comm. Pure Appl. Math.,43 (1990), 983–997.MathSciNetMATHGoogle Scholar
  4. [4]
    L.Bronsard and R.V.Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff., Eqns.,90 (1991) 211–237.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    P. Brunovský and B. Fiedler, Connection orbits in scalar reaction diffusion equations, Dynamics Reported vol1 (1988), 57–90, Wiley.Google Scholar
  6. [6]
    G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal.,92 (1986), 205–245.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    G. Caginalp and Y. Nishiura, The existence of travelling waves for phase field equations and convergence to sharp interface models in the singular limit, Quarterly of Appl. Math.,49 (1) (1991), 147–162.MathSciNetMATHGoogle Scholar
  8. [8]
    J. Carr, M. E. Gurtin, and M. Slemrod, Structured phase transitions on a finite interval, Arch. Rat. Mech. Anal.,86 (1984), 317–351.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    J. Carr and R. L. Pego, Metastable patterns in solutions ofut = ε2uxxf(u), Comm. Pure Appl. Math.,42 (1989), 523–576.MathSciNetMATHGoogle Scholar
  10. [10]
    R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Eqns., 27 (1978), 266–273.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    X. -Y. Chen, Dynamics of interfaces in reaction diffusion systems, Hiroshima Math. J.,21 (1991), 47–83.MATHGoogle Scholar
  12. [12]
    Y. -G. Chen, Y. Giga, and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geometry33 (1991), 749–786.MathSciNetMATHGoogle Scholar
  13. [13]
    S. -N. Chow, B. Deng, and D. Terman, The bifurcation of homoclinic and periodic orbits from two heteroclinic orbits, SIAM J. Math. Anal.,21 (1990), 179–204.MathSciNetMATHGoogle Scholar
  14. [14]
    B. Deng, The bifurcations of countable connections from a twisted heteroclinic loop, SIAM J. Math. Anal.22 (1991), 653–679.MATHGoogle Scholar
  15. [15]
    L. C. Evans and J. Spruck, Motion of level sets by mean curvature, J. Differential Geometry33 (1991), 601 - 633.MathSciNetGoogle Scholar
  16. [16]
    P. C. Fife, Boundary and interior transition layer phenomena for pairs of second order differential equations, J. Math. Anal.,54 (1976), 497–521.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    P. C. Fife, Propagator-controller systems and chemical patterns, inNonequilibrium Dynamics in Chemical Systems, A. Pacault and C. Vidal, eds., Springer-Verlag (1984), 76–88.Google Scholar
  18. [18]
    P. C. Fife, Understanding the patterns in the BZ reagent, J. Statist. Phys.,39 (1985), 687–703.MathSciNetCrossRefGoogle Scholar
  19. [19]
    P. C. Fife,Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics #53, SIAM, Philadelphia (1988).Google Scholar
  20. [20]
    P. C. Fife, Pattern dynamics for parabolic PDE’s, to appear in the Proc. of IMA Conference “Introduction to dynamical systems”, Sept. 1989.Google Scholar
  21. [21]
    P. C. Fife and W. M. Greenlee, Interior transition layers for elliptic boundary value problems with a small parameter, Russian Math. Surveys,29: 4 (1974), 103–131.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rat. Mech. Anal.,65 (1977), 335–361.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    H. Fujii and Y. Hosono, Neumann layer phenomena in nonlinear diffusion systems,Recent Topics in Nonlinear PDE, M. Mimura and T. Nishida, Math. Studies, 98, North-Holland, Amsterdam (1983), 21–38.Google Scholar
  24. [24]
    H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Phisica5D (1982), 1–42.MathSciNetGoogle Scholar
  25. [25]
    H. Fujii, Y. Nishiura and Y. Hosono, On the structure of multuple existence of stable stationary solutions in systems of reaction- diffusion equations, Patterns and Waves - Qualitative Analysis of Nonlinear Differential Equations, T. Nishida, M. Mimura and H. Fujii, Stud. Math. Appl.,18 (1986), 157–219.Google Scholar
  26. [26]
    G. Fusco and J. K. Hale, Slow-motion manifolds, dormant Instability, and singular perturbations, J. Dynamics and Diff. Eqns., 1 (1989), 75–94.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    R. Gardner and C. K. R. T Jones, A stability index for steady state solutions of boundary value problems for parabolic systems, J. Diff. Eqns.,91 (1991), 181–203.MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    J. K. Hale and K. Sakamoto, Existence and stability of transition layers, Japan J. Appl. Math.,5 (1988), 367–405.MathSciNetMATHGoogle Scholar
  29. [29]
    D. Henry,Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, New York, 1981.Google Scholar
  30. [30]
    D. Hilhorst, Y. Nishiura, and M. Mimura, A free boundary problem arising in some reacting-diffusing system, Proc. Roy. Soc. Edinburgh,118A (1991), 355–378.MathSciNetMATHGoogle Scholar
  31. [31]
    T. Ikeda and Y. Nishiura, Pattern selection for two breathers, SLAM Appl. Math., in press.Google Scholar
  32. [32]
    M. Ito, A remark on singular perturbation methods, Hiroshima Math. J.,14 (1985), 619–629.MATHGoogle Scholar
  33. [33]
    J. P. Keener and J. J. Tyson, Spiral waves in the Belousov- Zhabotinsky reaction, Physica,21D (1986), 307–324.MathSciNetMATHGoogle Scholar
  34. [34]
    H. Kokubu, Homoclinic and heteroclinic bifurcation of vector fields, Japan J. Appl. Math.,5 (1987), 455–501.Google Scholar
  35. [35]
    H. Kokubu, Y. Nishiura, and H. Oka, Heteroclinic and homoclinic bifurcation in bistable reaction diffusion systems, J. Diff. Eqns.,86 (2) (1990), 260–341.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    J. S. Langer, Instabilities and pattern formation in crystal growth, Review of Modern Physics,52 (1) (1980), 1–28.CrossRefGoogle Scholar
  37. [37]
    H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. RIMS, Kyoto Univ.,15 (1979), 401–454.Google Scholar
  38. [38]
    H. Meinhardt,Models of biological pattern formation. Academic Press, 1982.Google Scholar
  39. [39]
    M. Mimura, M. Tabata, and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal.,11 (1980), 613–631.MathSciNetMATHGoogle Scholar
  40. [40]
    J. D. Murray,Mathematical Biology. Springer-Verlag, 1989.Google Scholar
  41. [41]
    Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal.,13 (1982), 555–593.MathSciNetMATHGoogle Scholar
  42. [42]
    Y. Nishiura, Singular limit approach to stability and bifurcation for bistable reaction diffusion systems. In “Proceeding of the Workshop on Nonlinear PDE’s, Provo, Utah, March 1987” (P. Bates and P. Fife,), Rocky Mountain J. Math.,21 (2) (1991), 727–767.Google Scholar
  43. [43]
    Y. Nishiura, Singular limit eigenvalue problem in higher dimensional space, Proceeding of International Symposium on Functional Differential Equations and Related Topics, Kyoto, Japan, 30 August - 2 September, T. Yoshizawa and J. Kato, World Scientific, 1991.Google Scholar
  44. [44]
    Y. Nishiura and H. Fujii, Stability of singularly perturbed solutions to systems of reaction diffusion equations, SIAM J. Math. Anal.,18 (1987), 1726–1770.MathSciNetMATHGoogle Scholar
  45. [45]
    Y. Nishiura and H. Fujii, SLEP method to the stability of singularly perturbed solutions with multiple internal transition layers in reaction-diffusion systems, Proc. NATO Workshop “Dynamics of Infinite Dimensional Systems”, J. Hale and S.N. Chow, NATO ASI SeriesF37 (1986), 211–230.Google Scholar
  46. [46]
    Y. Nishiura and M. Mimura, Layer oscillations in reaction- diffusion systems, SIAM J. Appl. Math.,49 (1989), 481–514.MathSciNetMATHGoogle Scholar
  47. [47]
    Y. Nishiura, M. Mimura, H. Ikeda, and H. Fujii, Singilar limit analysis of stability of travelling wave solutions in bistable reaction-diffusion systems, SIAM J. Math. Anal.,21 (1990), 85–122.MathSciNetMATHGoogle Scholar
  48. [48]
    Y. Nishiura and H. Suzuki, Coalescence,repulsion, and nonuniqueness for singular limit reaction diffusion systems, to appear in the Proceedings of the International Taniguchi Symposium on Nonlinear Partial Differential Equations and Applications, Katata, 23 August - 29 August, 1989 ( Eds. T.Nishida and M.Mimura).Google Scholar
  49. [49]
    Y. Nishiura and H. Suzuki, Stability of generic interface of reaction diffusion systems in higher space dimensions, in preparation.Google Scholar
  50. [50]
    Y. Nishiura and T. Tsujikawa, Instability of singularly perturbed Neumann layer solutions in reaction-diffusion systems, Hiroshima Math. J.,20 (1990), 297–329.MathSciNetMATHGoogle Scholar
  51. [51]
    T. Ohta and M. Mimura, Pattern dynamics in excitable reaction- diffusion media,Formation, Dynamics and Stabilities of Patterns, K. Kawasaki, M. Suzuki, and A. Onuki, 1, World Scientific, 1990.Google Scholar
  52. [52]
    T. Ohta, M. Mimura, and R. Kobayashi, Higher-dimensional localized pattern in excitable media, Physics 34D (1989), 115–144.MathSciNetMATHGoogle Scholar
  53. [53]
    P. Pelce (editor),Dynamics of curved fronts, Perspective in Physics, Academic Press, 1988.Google Scholar
  54. [54]
    J. Rinzel and D. Terman, Propagation phenomena in a bistable reaction-diffusion systems, SIAM J. Appl. Math., 42 (1982) 1111–1137.MathSciNetMATHGoogle Scholar
  55. [55]
    K. Sakamoto, Construction and stability analysis of transition layer solutions in reaction-diffusion systems, Tohoku Math. J.,42 (1990), 17–44.MathSciNetMATHGoogle Scholar
  56. [56]
    H. Suzuki, Y. Nishiura, and H. Ikeda, Stability of traveling waves and a relation between the Evans function and the SLEP equation, submitted for publication.Google Scholar
  57. [57]
    M. Taniguchi and Y. Nishiura, Instability of planar interfaces in reaction diffusion systems, SIAM Math. Anal., in press.Google Scholar
  58. [58]
    A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond.,B237 (1952), 37–72.CrossRefGoogle Scholar
  59. [59]
    R. S. Varga,Matrix iterative analysis, Prentice-Hall, 1962.Google Scholar
  60. [60]
    J. H. Wilkinson,The algebraic eigenvalue problem, Oxford, Clarendon Press, 1965.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Yasumasa Nishiura
    • 1
  1. 1.Division of Mathematics and Informatics, Faculty of Integrated Arts and SciencesHiroshima UniversityHigashi-HiroshimaJapan

Personalised recommendations