论文标题
自主/周期性抛物线方程的非随机点
Non-wandering points for autonomous/periodic parabolic equations on the circle
论文作者
论文摘要
我们研究以下标量反应扩散方程在圆圈$ s^1 $，\ begin {equation*} u_ {t} = u_ {xx}+f（t，u，u _ {x}），\，\，\，\，t> 0， s^{1} = \ mathbb {r}/2π\ mathbb {z}，\ end {equation*}，其中$ f $独立于$ t $或$ t $ - periodic in $ t $。假设该方程式承认一个紧凑的全球吸引子。事实证明，任何非随机点都是系统的限制点（也就是说，这是大约$ω$ limit设置的一个点）。更确切地说，在自主情况下，证明任何非随机点都是固定点，要么在圆上产生旋转波。在周期性的情况下，证明任何非随机点都是周期点或在圆环上产生旋转波。 特别是，如果$ f（t，u，-u_x）= f（t，u，u_x）$，则任何非随机点是自主情况下的固定点，并且是周期性情况下的周期性点。
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}/2π\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 $ω$-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.
