We consider the parabolic problem ut -Δu = h(t)f (u) in Ω×(0, T) with a Dirichlet condition on the boundary and f, h ∈ C[0, ∞). The initial data is assumed in the space {u0 ∈ C0 (Ω); u0 ≥ 0}, where Ω is a either bounded or unbounded domain. We find conditions that guarantee the global existence (or the blow up in finite time) of nonnegative solutions. © 2014 Texas State University - San Marcos.