  In this paper, we consider the following nonlocal parabolic equation
  \begin{equation*}
  u_{t}-\Delta u=\left(
\int_{\Omega}\frac{|u(y,t)|^{2^{\ast}_{\mu}}}{|x-y|^{\mu}}dy\right)
|u|^{2^{\ast}_{\mu}-2}u,\ \text{in}\ \Omega\times(0,\infty),
  \end{equation*}
  where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $0<\mu<N$ and
$2^{\ast}_{\mu}=(2N-\mu)/(N-2)$ denotes the critical exponent in the sense of
the Hardy-Littlewood-Sobolev inequality. We first introduce the stable and
unstable sets for the equation and prove that the problem has a potential well
structure. Next, we investigate the global asymptotic behavior of the
solutions.
  In particular, we study the behavior of the global solutions that intersect
neither with the stable set nor the unstable set.
  Finally, we prove that global solutions have $L^{\infty}$-uniform bound under
some natural conditions.
