WIAS Preprint No. 599, (2000)

A sequence of order relations, encoding heteroclinic connections in scalar parabolic PDE

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Authors

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

2010 Mathematics Subject Classification

• 35K57 37L30 35B41 34C37

Keywords

• calar semilinear parabolic PDE, order structures; attractors, heteroclinic connections, meandric permutations, nodal properties

DOI

10.20347/WIAS.PREPRINT.599

Abstract

We address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations

u_t = u_xx + ƒ (x, u, u_x), 0 < x < 1

on a bounded interval with Neumann conditions. Introducing a sequence of order relations, we prove a new and simple criterion for the existence of heteroclinic connections, using only information about nodal properties of solutions to the stationary ODE problem. This result allows also for a complete classiffication of possible attractors in terms of the permutation of the equilibria, given by their order at the two boundaries of the interval.

Appeared in

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

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