  Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)}
\qquad \begin{cases} \partial_t u=D_1\Delta u+v^p\quad &
\quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2\Delta v+u^q\quad &
\quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ u,v\ge 0 & \quad\mbox{in}\quad{\bf
R}^N\times(0,T),\\ (u(\cdot,0),v(\cdot,0))=(\mu,\nu) & \quad\mbox{in}\quad{\bf
R}^N, \end{cases} \] where $N\ge 1$, $T>0$, $D_1>0$, $D_2>0$, $0<p\le q$ with
$pq>1$ and $(\mu,\nu)$ is a pair of Radon measures or nonnegative measurable
functions in ${\bf R}^N$. In this paper we study qualitative properties of the
initial trace of the solution $(u,v)$ and obtain necessary conditions on the
initial data $(\mu,\nu)$ for the existence of solutions to problem (P).
