Connecting orbits in scalar reaction diffusion equations

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


We consider the flow of a one-dimensional reaction diffusion equation
$$ {u_t} = {u_{xx}} + f(u),x \in (0,1)$$
with Dirichlet boundary conditions
$$ u(t,0) = u(t,1) = 0{\text{ }}$$
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
$$ \mathop {\lim }\limits_{t \to - \infty } \;u\left( {t, \cdot } \right) = \upsilon \;\mathop {\lim }\limits_{t \to - \infty } \;u\left( {t, \cdot } \right) = w$$
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?


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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

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