EXISTENCE AND NON-EXISTENCE OF SOLUTIONS FOR HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA

We establish the existence, non-existence and uniqueness of the local solutions of the Hardy parabolic equation u_t − ∆u = h(t)| · |^−γ g(u) on Ω × (0, T) with Dirichlet boundary conditions. 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 data have a singularity at the origin.