Non-wandering points for autonomous/periodic parabolic equations on the circle

TL;DR

This paper investigates the structure of non-wandering points in scalar reaction-diffusion equations on the circle, revealing they are either fixed points, periodic points, or rotating waves, depending on the system's time dependence.

Contribution

It characterizes non-wandering points for autonomous and periodic parabolic equations on the circle, showing they are limit points such as fixed points, periodic points, or rotating waves.

Findings

Non-wandering points are limit points in the system's omega-limit set.

In autonomous systems, non-wandering points are fixed points or rotating waves.

In periodic systems, non-wandering points are periodic points or rotating waves.

Abstract

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle $S^1$, \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2\pi \mathbb{Z}, \end{equation*} where $f$ is independent of $t$ or $T$-periodic in $t$. Assume that the equation admits a compact global attractor. It is proved that, any non-wandering point is a limit point of the system (that is, it is a point in some $\omega$-limit set). More precisely, in the autonomous case, it is proved that any non-wandering point is either a fixed point or generates a rotating wave on the circle. In the periodic case, it is proved that any non-wandering point is a periodic point or generates a rotating wave on a torus. In particular, if $f(t,u,-u_x)=f(t,u,u_x)$, then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case.