We consider the parabolic problem ut−∆u = 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.