We study the behaviour of the solutions to the Cauchy problem {ρ (x) ut=∆ u+ up, x∈ RN, t∈(0, T), u (x, 0)= u0 (x), x∈ RN, with p> 0 and a positive density ρ satisfying ρ (x)∼| x|− σ for large| x|, 0< σ< 2< N. We consider both the cases of a bounded density and the singular density ρ (x)=| x|− σ. We are interested in describing sharp decay conditions on the data at infinity that guarantee local/global existence of solutions, which are unique in classes of functions with the same decay. We prove that larger data give rise to instantaneous complete blow-up. We also deal with the occurrence of finite-time blow-up. We prove that the global existence exponent is p0= 1, while the Fujita exponent depends on σ, namely pc= 1+ 2