Describing a class of global attractors via symbol sequences

Jörg Härterich; Matthias Wolfrum

doi:10.3934/dcds.2005.12.531

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2005, Volume 12, Issue 3: 531-554. Doi: 10.3934/dcds.2005.12.531

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Describing a class of global attractors via symbol sequences

Jörg Härterich^1, and
Matthias Wolfrum^2,

1.
Institut für Mathematik I, Freie Universität Berlin, Arnimallee 2-6, 14195 Berlin
2.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin

Received: April 30, 2003

Revised: September 30, 2004

Published: March 2005

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Abstract

We study a singularly perturbed scalar reaction-diffusion equation on a bounded interval with a spatially inhomogeneous bistable nonlinearity. For certain nonlinearities, which are piecewise constant in space on $k$ subintervals, it is possible to characterize all stationary solutions for small $\varepsilon$ by means of sequences of $k$ symbols, indicating the behavior of the solution in each subinterval. Determining also Morse indices and zero numbers of the equilibria in terms of the symbol sequences, we are able to give a criterion for heteroclinic connections and a description of the associated global attractor for all $k$.

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Mathematics Subject Classification: 35B40, 35B41, 35K57, 34E15, 37C29, 37L30.

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