WIAS Report No. 15, (1998)

Geometry of heteroclinic cascades in scalar parabolic differential equations


  • Wolfrum, Matthias
    ORCID: 0000-0002-4278-2675

2010 Mathematics Subject Classification

  • 35K57 35B05 58F12 58F39


  • Reaction-diffusion equations, global attractor, Morse-Smale systems, heteroclinic orbits, nodal properties, meandric permutations




We investigate the geometrical properties of the attractor for semilinear scalar parabolic PDEs on a bounded interval with Neumann boundary conditions. Using the nodal properties of the stationary solutions which are determined by an ordinary boundary value problem, we obtain crucial information about the long-time behavior for the full PDE. Especially, we prove a criterion for the intersection of strong- stable and unstable manifolds in the finite dimensional Morse-Smale flow on the attractor.

Appeared in

  • PDE, J. of Differential. Equations 183(2002), pp. 56-78.

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