Geometry of heteroclinic cascades in scalar parabolic differential equations
- Wolfrum, Matthias
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.
- PDE, J. of Differential. Equations 183(2002), pp. 56-78.