We consider the singular nonlinear equation ut−Δu=|⋅|−γf(u) in Ω×(0,T) with γ>0 and Dirichlet conditions on the boundary. This equation is known in the literature as a Hardy parabolic equation. The function f:[0,∞)→[0,∞) is continuous and non-decreasing, and Ω is either a smooth bounded domain containing the origin or the whole space RN. We determine necessary and sufficient conditions for the existence and non-existence of solutions for initial data u0∈Lr(Ω),u0≥0, with 1≤r<∞. We also give a uniqueness result.