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A Poincaré-Bendixson theorem for scalar reaction diffusion equations. (English) Zbl 0704.35070
Authors’ summary: “For scalar equations \(u_ t=u_{xx}+f(x,u,u_ x)\) with \(x\in S^ 1\) and \(f\in C^ 2\) we show that the classical theorem of Poincaré and Bendixson holds: the \(\omega\)-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.”
Reviewer: B.D.Sleeman

MSC:
35K57 Reaction-diffusion equations
35B99 Qualitative properties of solutions to partial differential equations
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