In this study, we consider the parabolic systems ut-δu=f(t)vp, vt-δv=f(t)uq and ut-δu=f(t)(ur+vp), vt-δv=f(t)(uq+vs) in Ω×(0, T) with homogeneous Dirichlet boundary condition, p, q, r, s≥1 and f∈C[0, ∞). The initial data are considered in the space {(u0,v0)∈[C0(Ω)]2;u0,v0≥0}, where Ω is a general domain (bounded or unbounded) with a smooth boundary. We find the conditions that guarantee the global existence (or the blow-up in finite time) of nonnegative solutions. These conditions are given in terms of the asymptotic behavior of the solution of the homogeneous linear problem: ut-δu=0 in Ω×(0, ∞).