  In this paper, we consider the following non-local semi-linear parabolic
equation with advection: for $1 \le p<1+\frac{2}{N}$, \begin{equation*}
\begin{cases} u_t+v \cdot \nabla u-\Delta u=|u|^p-\int_{\mathbb T^N} |u|^p
\quad & \textrm{on} \quad \mathbb T^N, \\ \\ u \ \textrm{periodic} \quad &
\textrm{on} \quad \partial \mathbb T^N \end{cases} \end{equation*} with initial
data $u_0$ defined on $\mathbb T^N$. Here $v$ is an incompressible flow, and
$\mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension.
  We first prove the local existence of mild solutions to the above equation
for arbitrary data in $L^2$. We then study the global existence of the
solutions under the following two scenarios: (1). when $v$ is a mixing flow;
(2). when $v$ is a shear flow. More precisely, we show that under these
assumptions, there exists a global solution to the above equation in the sense
of $L^2$.
