Advertisement

Connecting orbits in scalar reaction diffusion equations

  • P. Brunovský
  • B. Fiedler
Chapter
Part of the Dynamics Reported book series (DYNAMICS, volume 1)

Abstract

We consider the flow of a one-dimensional reaction diffusion equation
$$ {u_t} = {u_{xx}} + f(u),x \in (0,1)$$
(1.1)
with Dirichlet boundary conditions
$$ u(t,0) = u(t,1) = 0{\text{ }}$$
(1.2)
Let v, w denote stationary, i.e. t-independent solutions. We say that v connects to w, if there exists an orbit u(t, x) of (1.1), (1.2) such that
$$ \mathop {\lim }\limits_{t \to - \infty } \;u\left( {t, \cdot } \right) = \upsilon \;\mathop {\lim }\limits_{t \to - \infty } \;u\left( {t, \cdot } \right) = w$$
(1.3)
i.e. u(t, ·) is a heteroclinic orbit connecting v to w. In this report we address the following question:

(*) Given v, which stationary solutions w does it connect to?

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Angenent, The Morse–Smale property for a semilinear parabolic equation, to appear in J. Diff. Eq. 62 (1986), 427–442.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    F. V. Atkinson, Discrete and Continuous Boundary Problems. Academic Press, London, 1964.zbMATHGoogle Scholar
  3. [3]
    G. Birkhoff and G.-C. Rota, Ordinary Differential Equations. Ginn & Co, Boston, 1959.Google Scholar
  4. [4]
    P. Brunovskÿ and S.-N. Chow, Generic properties of stationary state solutions of reaction diffusion equations. J. Diff. Eq., 53 (1984), 1–23.zbMATHCrossRefGoogle Scholar
  5. [5]
    P. Brunovskÿ and B. Fiedler, Numbers of zeros on invariant manifolds in scalar reaction diffusion equations, Non!. Analysis 10 (1986), 179–193.zbMATHGoogle Scholar
  6. [6]
    P. Brunovskÿ and B. Fiedler, Simplicity of zeros in scalar parabolic equations, J. Diff. Eq. 62 (1986), 237–241.zbMATHCrossRefGoogle Scholar
  7. [7]
    P. Brunovskÿ and B Fiedler, Heteroclinic connections of stationary solutions of scalar reaction diffusion equations. Proc. Int. Stefan Banach Inst., B. Bojarski (ed.), Warszawa, 1984, to appear.Google Scholar
  8. [8]
    N. Chafee and E. Infante, A bifurcation problem for a nonlinear parabolic equation. J. Appl. Anal., 4 (1974), 17–37.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    C. C. Conley, Isolated invariant sets and the Morse index. Conf. Board Math. Sci., 38, AMS, Providence, 1978.Google Scholar
  10. [10]
    C. C. Conley and J. Smoller, Topological techniques in reaction diffusion equations, in Biologa! Growth and Spread, Proc., Heidelberg, 1979, Jäger, Rost, Tautu (eds), Springer Lect. Notes Biomath., 38, 473–83.MathSciNetGoogle Scholar
  11. [11]
    C. C. Conley and J. Smoller, Birfurcation and stability of stationary solutions of the Fitz–Hugh–Nagumo equations, preprint, 1984.Google Scholar
  12. [12]
    A. Dold, Lectures on Algebraic Topology. Springer, Heidelberg, 1972.zbMATHGoogle Scholar
  13. [13]
    I. Gelfand, Some problems in the theory of quasilinear equations. AMS Transl., ser. 2, 29 (1963), 295–381.Google Scholar
  14. [14]
    J. K. Hale, Infinite dimensional dynamical systems, in Geometric Dynamics, Proc. Rio De Janeiro 1981, J. Palis Jr. (ed.), Springer Lect. Notes Math. 1007, 379–400.Google Scholar
  15. [15]
    Ph. Hartman, Ordinary Differential Equations ( 2nd edn ). Birkhäuser, Boston, 1982.zbMATHGoogle Scholar
  16. [16]
    D. Henry, Geometric Theory of Semilinear Parabolic Equations. Springer Lect. Notes Math., New York, 1981.Google Scholar
  17. [17]
    D. Henry, Some infinite dimensional Morse—Smale systems defined by parabolic differential equations. J. Diff. Eq., 59 (1985), 165–205.zbMATHCrossRefGoogle Scholar
  18. [18]
    A. C. Hindmarsh, LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM-Signum Newsletter 15 (1980), 10–11.CrossRefGoogle Scholar
  19. [19]
    M. W. Hirsch, Differential Topology. Springer, New York, 1976.zbMATHCrossRefGoogle Scholar
  20. [20]
    H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation. J. Fac. Sci. Univ. Tokyo Sec. IA, 29 (1982), 401–41.MathSciNetzbMATHGoogle Scholar
  21. [21]
    D. Hart, A. C. Lazer and P. J. McKenna, Multiple solutions of two point boundary value problems with jumping nonlinearities, to appear in J. Diff. Eq.Google Scholar
  22. [22]
    J. Mallet-Paret, Morse decompositions and global continuation of periodic solutions for singularly perturbed delay equations, in Systems of Nonlinear Partial Differential Equations, J. M. Ball (ed.), D. Reidel, Dordrecht, 1983, 351–66.CrossRefGoogle Scholar
  23. K. Nickel, Gestaltaussagen über Lösungen parabolischer Differentialgleichungen. Crelles J. Reine Angew. Math., 211 (1962), 78–94.Google Scholar
  24. [24]
    P. Polâeik, Generic bifurcation of stationary solutions of the Neumann problem for reaction diffusion equations. Thesis, Bratislava, 1984.Google Scholar
  25. [25]
    M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, 1967.Google Scholar
  26. [26]
    R. Schaaf, Global solution branches via time-maps, preprint 1986.Google Scholar
  27. [27]
    J Smoller, Shock Waves and Reaction Diffusion Equations. Springer, New York, 1983.zbMATHCrossRefGoogle Scholar
  28. [28]
    J. A. Smoller and J. S. Shi, Analytical and topological methods for reaction diffusion equations, in Modelling of Patterns in Space and Time, Proc. Heidelberg, 1983, Jäger, Murray (eds), Springer Lect. Notes Biomath., 55, 350–63.MathSciNetGoogle Scholar
  29. [29]
    J. Smoller, A. Tromba and A. Wasserman, Nondegenerate solutions of boundary value problems. Nonl. Analysis, 4 (1980), 207–215.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    J. Smoller and A. Wasserman, Generic bifurcations of steady state solutions. J. Diff. Eq., 52 (1984), 432–8.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    J. Smoller and A. Wasserman, Global bifurcation of steady state solutions. J. Diff. Eq., 39 (1981), 269–90.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    W. Walter, Differential and Integral Inequalities. Springer, Heidelberg, 1970.zbMATHCrossRefGoogle Scholar

Copyright information

© John Wiley & Sons and B. G. Teubner 1988

Authors and Affiliations

  • P. Brunovský
    • 1
  • B. Fiedler
    • 2
  1. 1.Universita KomenskéhoBratislavaSlovakia
  2. 2.Institut für Angewandte MathematikHeidelbergGermany

Personalised recommendations