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

Abstract

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle S 1 , u t = u x x + f ( t , u , u x ) , t > 0 , x ∈ S 1 = R / 2 π Z , 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 ω -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.