  Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic
system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad &
$\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta
v+u^q\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\
(u(\cdot,0),v(\cdot,0))=(\mu,\nu) & $\quad\mbox{in}\quad{\mathbb{R}}^N,$ } \]
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 nonnegative Radon measures or locally integrable
nonnegative functions in ${\mathbb R}^N$. In this paper we establish sharp
sufficient conditions on the initial data for the existence of solutions to
problem~(P) using uniformly local Morrey spaces and uniformly local weak
Zygmund type spaces.
