Almost Automorphically and Almost Periodically Forced Circle Flows of Almost Periodic Parabolic Equations on S^1

Abstract

We consider the skew-product semiflow which is generated by a scalar reaction-diffusion equation \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 uniformly almost periodic in $t$. The structure of the minimal set $M$ is thoroughly investigated under the assumption that the center space $V^c(M)$ associated with $M$ is no more than $2$-dimensional. Such situation naturally occurs while, for instance, $M$ is hyperbolic or uniquely ergodic. It is shown in this paper that $M$ is a $1$-cover of the hull $H(f)$ provided that $M$ is hyperbolic (equivalently, ${\rm dim}V^c(M)=0$). If ${\rm dim}V^c(M)=1$ (resp. ${\rm dim}V^c(M)=2$ with ${\rm dim}V^u(M)$ being odd), then either $M$ is an almost $1$-cover of $H(f)$ and topologically conjugate to a minimal flow in $\mathbb{R}\times H(f)$; or $M$ can be (resp. residually) embedded into an almost periodically (resp. almost automorphically) forced circle-flow $S^1\times H(f)$. When $f(t,u,u_x)=f(t,u,-u_x)$ (which includes the case $f=f(t,u)$), it is proved that any minimal set $M$ is an almost $1$-cover of $H(f)$. In particular, any hyperbolic minimal set $M$ is a $1$-cover of $H(f)$. Furthermore, if ${\rm dim}V^c(M)=1$, then $M$ is either a $1$-cover of $H(f)$ or is topologically conjugate to a minimal flow in $\mathbb{R}\times H(f)$. For the general spatially-dependent nonlinearity $f=f(t,x,u,u_{x})$, we show that any stable or linearly stable minimal invariant set $M$ is residually embedded into $\mathbb{R}^2\times H(f)$.