Existence and non-existence of solutions for hard parabolic equations with singular initial data

Abstract

We establish the existence, non-existence and uniqueness of the local solutions ofthe Hardy parabolic equationut−∆u=h(t)|·|−γg(u) on Ω×(0,T) with Dirichlet boundaryconditions. We assume that Ω with 0∈Ω is a smooth domain bounded or unbounded,h∈C(0,∞),g∈C([0,∞)) is a non-decreasing function, 0< γ <min{2,N}, and the initial datahave a singularity at the origin.