Archive for Rational Mechanics and Analysis

, Volume 107, Issue 4, pp 325–345

A Poincaré-Bendixson theorem for scalar reaction diffusion equations

  • Bernold Fiedler
  • John Mallet-Paret

DOI: 10.1007/BF00251553

Cite this article as:
Fiedler, B. & Mallet-Paret, J. Arch. Rational Mech. Anal. (1989) 107: 325. doi:10.1007/BF00251553


For scalar equations
$$u_t = u_{xx} + f(x, u, u_x )$$
with x ε S1 and f ε C2 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.

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Bernold Fiedler
    • 1
    • 2
  • John Mallet-Paret
    • 1
    • 2
  1. 1.Institute of Applied MathematicsUniversity of HeidelbergWest Germany
  2. 2.Division of Applied MathematicsBrown UniversityProvidence