论文标题
几乎是$ s^1 $或$ \ mathbb {r} $在一维几乎是周期性半线性热方程中的几乎自动派强制流动
Almost automorphically forced flows on $S^1$ or $\mathbb{R}$ in one-dimensional almost periodic semilinear heat equations
论文作者
论文摘要
In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L带有周期性边界条件\ begin {equation} \ label {bdc1} u（t，0）= u（t，t，l），\ quad u_x（t，0）= u_x（t，t，t，l），\ end end eND {equation}，其中$ f $在$ T $中均匀定期。特别是，我们研究了偏斜产品半节的极限集的拓扑结构。事实证明，任何紧凑的最小不变集（在本文中，我们都将其称为最小集合）都可以将残留嵌入到一个不变的一组中，这些集合几乎是几乎是自动形态的流量，$ s^1 = \ mathbb {r}/l}/l}/l \ mathbb {z} $。特别是，如果$ f（t，u，p）= f（t，u，-p）$，则最小设置上的流量从拓扑结合到几乎定期造成的最小流量上的$ \ mathbb {r} $。此外，事实证明，任何有限轨道的$ω$限制集都包含在最小两个集合中，这些集合无法通过相位翻译互相获得。 此外，我们进一步考虑了\ eqref {eq0}的偏压产品半流的渐近动力学，并具有neumann边界条件 \ begin {equation*} \ label {bcd2} u_x（t，0）= u_x（t，l）= 0， \ end {equation*} 或Dirichlet边界条件 \ begin {equation*} \ label {bdc3} u（t，0）= u（t，l）= 0。 \ end {equation*}在$ f $的某些直接假设下，在本文中证明，在任何最小的\ eqref {eq0}集中流动，具有neumann边界条件或dirichlet边界条件，拓扑结合在$ \ m naterbbbb {r} $上几乎定期定期结合。 最后，给出反例以表明即使对于准周期方程，我们在这里获得的结果总体上也无法进一步改善。
In this paper, we consider the asymptotic dynamics of the skew-product semiflow generated by the following time almost-periodically forced scalar reaction-diffusion equation \begin{equation}\label{eq0} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\, 0<x<L \end{equation} with periodic boundary condition \begin{equation} \label{bdc1} u(t,0)=u(t,L),\quad u_x(t,0)=u_x(t,L), \end{equation} where $f$ is uniformly almost periodic in $t$. In particular, we study the topological structure of the limit sets of the skew-product semiflow. It is proved that any compact minimal invariant set (throughout this paper, we refer to it as a minimal set) can be residually embedded into an invariant set of some almost automorphically-forced flow on a circle $S^1=\mathbb{R}/L\mathbb{Z}$. Particularly, if $f(t,u,p)=f(t,u,-p)$, then the flow on a minimal set topologically conjugates to an almost periodically-forced minimal flow on $\mathbb{R}$. Moreover, it is proved that the $ω$-limit set of any bounded orbit contains at most two minimal sets that cannot be obtained from each other by phase translation. In addition, we further consider the asymptotic dynamics of the skew-product semiflow generated by \eqref{eq0} with Neumann boundary condition \begin{equation*} \label{bcd2} u_x(t,0)=u_x(t,L)=0, \end{equation*} or Dirichlet boundary condition \begin{equation*}\label{bdc3} u(t,0)=u(t,L)=0. \end{equation*} Under certain direct assumptions on $f$, it is proved in this paper that the flow on any minimal set of \eqref{eq0}, with Neumann boundary condition or Dirichlet boundary condition, topologically conjugates to an almost periodically-forced minimal flow on $\mathbb{R}$. Finally, a counterexample is given to show that even for quasi-periodic equations, the results we obtain here cannot be further improved in general.
