A Poincaré-Bendixson theorem for scalar reaction diffusion equations. (English) Zbl 0704.35070

Fiedler, Bernold; Mallet-Paret, John

Arch. Ration. Mech. Anal. 107, No. 4, 325-345 (1989).

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

Keywords:

Poincaré-Bendixson theorem; reaction diffusion equations

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