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A Poincaré-Bendixson theorem for scalar reaction diffusion equations

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Abstract

For scalar equations

$$u_t = u_{xx} + f(x, u, u_x )$$

with x ε S 1 and f ε C 2 we show that the classical theorem of Poincaré and Bendixson holds: the ω-limit set of any bounded solution satisfies exactly one of the following alternatives:

  • - it consists in precisely one periodic solution, or

  • - it consists of solutions tending to equilibrium as \(t \to \pm \infty \)

This is surprising, because the system is genuinely infinite-dimensional.

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Authors and Affiliations

  1. Institute of Applied Mathematics, University of Heidelberg, West Germany

    Bernold Fiedler & John Mallet-Paret

  2. Division of Applied Mathematics, Brown University, Providence, R.I.

    Bernold Fiedler & John Mallet-Paret

Authors
  1. Bernold Fiedler
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  2. John Mallet-Paret
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Communicated by K. Kirchgässner

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Fiedler, B., Mallet-Paret, J. A Poincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Rational Mech. Anal. 107, 325–345 (1989). https://doi.org/10.1007/BF00251553

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  • DOI: https://doi.org/10.1007/BF00251553

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