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Non-wandering points for autonomous/periodic parabolic equations on the circle
Journal of Differential Equations ( IF 2.4 ) Pub Date : 2021-06-26 , DOI: 10.1016/j.jde.2021.06.023
Wenxian Shen , Yi Wang , Dun Zhou

We study the properties of non-wandering points of the following scalar reaction-diffusion equation on the circle S1,ut=uxx+f(t,u,ux),t>0,xS1=R/2πZ, 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,ux)=f(t,u,ux), then any non-wandering point is a fixed point in the autonomous case, and is a periodic point in the periodic case.



中文翻译:

圆上自主/周期抛物线方程的非游移点

我们研究下列标量反应扩散方程在圆上的非游移点的性质 1,=XX+F(,,X),>0,X1=电阻/2πZ,其中˚F是独立的Ť中为周期。假设方程允许一个紧凑的全局吸引子。证明,任何非游移点都是系统的一个极限点(即它是某个ω-极限集中的一个点)。更准确地说,在自治的情况下,证明了任何非漂移点要么是固定点,要么在圆上产生旋转波。在周期情况下,证明任何非游移点都是周期点或在环面上产生旋转波。特别地,如果F(,,-X)=F(,,X),则任意非游移点在自治情况下为不动点,在周期情况下为周期点。

更新日期:2021-06-28
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