On the local existence for Hardy parabolic equations with singular initial data

We consider the singular nonlinear equation u(t) - Delta u = vertical bar.vertical bar(-gamma) f(u) in Omega x (0, T) with gamma > 0 and Dirichlet conditions on the boundary. This equation is known in the literature as a Hardy parabolic equation. The function f : [0, infinity) -> [0, infinity) is continuous and non-decreasing, and Omega is either a smooth bounded domain containing the origin or the whole space R-N. We determine necessary and sufficient conditions for the existence and non-existence of solutions for initial data u(0) is an element of L-r(Omega), u(0) >= 0, with 1 <= r < infinity. We also give a uniqueness result. (C) 2022 Elsevier Inc. All rights reserved.