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

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, $\mathrm{dim}V^c(M)=0$). If $\mathrm{dim}V^c(M)=1$ (resp. $\mathrm{dim}V^c(M)=2$ with $\mathrm{dim}V^u(M)$ 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 $\mathrm{dim}V^c(M)=1$, 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 ${\mathbb {R}}^2\times H(f)$.