A semilinear parabolic problem with variable reaction on a general domain

We are concerned with the parabolic equation ut−Δu=f(t)up(x) in Ω×(0,T) with homogeneous Dirichlet boundary condition, p∈C(Ω), f∈C([0,∞)) and Ω is either a bounded or an unbounded domain. The initial data is considered in the space {u0∈C0(Ω);u0≥0}. We find conditions that guarantee the global existence and 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.