  In this paper we will see that the global or local existence of solutions to
\begin{eqnarray*} \dfrac{\partial u_{1}}{\partial t} & = & \mathit{k}_{1} (t)
\Delta u_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial
u_{2}}{\partial t} & = & \mathit{k}_{2} (t) \Delta u_{2} + h_{2}(t)
u_{2}^{p_{22}} u_{1}^{p_{21}}, \end{eqnarray*} depends on the initial datums
and the global or local existence of solutions to \begin{eqnarray*}
\dfrac{dy_{1}}{dt} & = & h_{1}(t) y_{1}^{p_{11}}(t) y_{2}^{p_{12}}(t),\\
\dfrac{dy_{2}}{dt} & = & h_{2}(t) y_{2}^{p_{22}}(t) y_{1}^{p_{21}}(t).
\end{eqnarray*} We also give some bounds for the maximal existence time of the
partial differential system. Moreover, if such existence time is finite and
$\min\{p_{11} + p_{12},p_{22}+p_{21}\} > 1$ then we will prove the partial
differential system has solutions that blows-up at space infinite.
