  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 $\omega$-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.

Dynamical Systems;Analysis of PDEs