  In this paper, we obtain the so-called Fujita exponent to the following
parabolic system with time-weighted sources and degenerate coefficients $
u_{t}- \mbox{div} ( \omega(x)\nabla { u} )= t^{r} v^{p} $ and $ v_{t}-
\mbox{div} ( \omega(x)\nabla {v} )= t^{s} u^{p}$ in $\mathbb{R}^{N} \times
(0,T)$ with initial data belonging to $ \left[L^\infty(\mathbb{R}^N)\right]^2.$
Where $p,q > 0$ with $ pq > 1$; $r,s>-1 $; and either $\omega(x) = | x_1|^{a},$
or $\omega(x) = | x |^{b}$ with $a,b > 0$.
