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Describing a class of global attractors via symbol sequences

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

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