Abstract
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 an exact criterion for the intersection of strong-stable and strong-unstable manifolds in the finite dimensional Morse-Smale flow on the attractor.
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Wolfrum, M. Geometry of Heteroclinic Cascades in Scalar Parabolic Differential Equations. Journal of Dynamics and Differential Equations 14, 207–241 (2002). https://doi.org/10.1023/A:1012967428328
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DOI: https://doi.org/10.1023/A:1012967428328