Catastrophe Theory Concepts for Ignition/Extinction Phenomena

  • D. Meinköhn
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 48)


This investigation centers on a class of semi-linear elliptic boundary value problems which are to model the physical behaviour of certain stationary reaction-diffusion systems for which ignition and extinction is known to occur. It is shown that the solutions may be represented by the points of a state surface in an appropriate finite-dimensional state space and hence may be classified according to the concepts of Catastrophe Theory via an investigation of the projection image of the state surface in a subspace spanned by the control parameters which form a subset of the state variables. Due to the existence of a distinguished control parameter (which is designated by λ and is known as the “Frank-Kamenetzkiparameter”), the state surface is given by an explicit expression for λ in terms of the cther state variables. With the help of the maximum principle for elliptic differential equations another surface is shown to exist which — in terms of λ — bounds the state surface from below. For this bounding surface, the projection image in the subspace of the control parameters furnishes a bifurcation set which provides lower bounds for the associated critical singularities of the exact problem.


Control Parameter Bifurcation Diagram Regular Solution Singular Solution Catastrophe Theory 
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  1. [1]
    R. Gilmore, Nucl.Phys.B (Proc.Suppl.) 2, 191(1987).CrossRefGoogle Scholar
  2. [2]
    W. Güttinger, p.23 in “Structural Stability in Physics”, W. Güttinger and H. Eikemeier (Eds.), Springer Ser.Synergetics Vol.4 (1979).Google Scholar
  3. [3]
    M.M. Vajnberg, W.A. Trenogin, “Theory of Branching Solutions of Non-Linear Equations”, Noordhoff, Leyden (1974).Google Scholar
  4. [4]
    R. Magnus, T. Poston, p.63 in “Structural Stability in Physics”, W.Güttinger and H. Eikemeier (Eds.), Springer Ser. Synergetics Vol.4 (1979).Google Scholar
  5. [5]
    R. Seydel, V. Hlavacek, Chem.Eng.Sci. 42, 1281(1987).Google Scholar
  6. [6]
    R. Aris, “The Mathematical Theory of Diffusion and Reaction”, Clarendon Press, Oxford (1975).Google Scholar
  7. [7]
    A. Friedman, “Partial Differential Equations of Parabolic type”, Prentice Hall, Englewood Cliffs (1964).MATHGoogle Scholar
  8. [8]
    I.M. Gelfand, Am.Math.Soc.Translations, Ser.2, 29, 295 (1963).Google Scholar
  9. [9]
    P.L. Lions, SIAM-Rev. 24, 441(1982).CrossRefGoogle Scholar
  10. [10]
    W.-M. Ni, L.A. Peletier, J. Serrin (Eds.), “Nonlinear Diffusion Equations and their Equilibrium States”, Springer, Berlin (1988).Google Scholar
  11. [11]
    D.D. Joseph, E.M. Sparrow, Quart. J. Appl. Math. 28, 329 (1970).MathSciNetGoogle Scholar
  12. [12]
    H.B. Keller, p.359 in “Applications of Bifurcation Theory”, P.H. Rabinowitz (Ed.), Academic Press, London (1977).Google Scholar
  13. [13]
    D.A. Frank-Kamenetzkii,“Diffusion and Heat Transfer in Chemical Kinetics”, Plenum Press, New York (1969).Google Scholar
  14. [14]
    D.H. Sattinger,“Topics in Stability and Bifurcation Theory”, Lecture Notes in Mathematics No.309, Springer, Berlin (1973).Google Scholar
  15. [15]
    D. Meinköhn, J. Chem. Phys. 74, 3603 (1981).ADSCrossRefGoogle Scholar
  16. [16]
    G. Nicolis, I. Prigogine,“Self-Organization in Nonequilibrium Systems”, Wiley, New York (1977).MATHGoogle Scholar
  17. [17]
    L.F. Razon, R.A. Schmitz, Chem.Eng.Sci. 42, 1005 (1987).CrossRefGoogle Scholar
  18. [18]
    D. Meinköhn, SIAM-J. Appl. Math. 48, 536 (1988).ADSCrossRefMATHMathSciNetGoogle Scholar
  19. [19]
    D. Meinköhn, SIAM-J. Appl. Math. 48, 792 (1988).CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    B. Gidas, W.-M. Ni, L. Nirenberg, Comm.Math.Phys. 68, 209 (1979).ADSCrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    D. Meinköhn, Chem.Eng.Sci., in press.Google Scholar
  22. [22]
    D. Meinköhn, Int.J. Heat Mass Transfer 23, 833 (1980).CrossRefMATHGoogle Scholar
  23. [23]
    W. Börsch-Supan, J. Appl. Math. Phys. 35, 332 (1984).CrossRefMATHGoogle Scholar
  24. [24]
    L.D. Landau, E.M. Lifshits, “Statistical Physics”, Pergamon, Oxford (1980).Google Scholar
  25. [25]
    D. Meinköhn, in:“Nonlinear Wave Processes in Excitable Media”, A.V. Holden, M. Markus, H.G. Othmer (Eds.), Plenum Press, in preparation.Google Scholar
  26. [26]
    K.H. Winters, K.A. Cliffe, Comb.Flame 62, 13 (1985).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • D. Meinköhn
    • 1
  1. 1.Fakultät für PhysikUniversität BielefeldBielefeld 1Fed. Rep. of Germany

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