The heat equation with singular nonlinearity and singular initial data

Abstract

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation ut - Δ u = a (x) uq + b (x) up in a bounded domain and with Dirichlet's condition on the boundary. We consider here a ∈ Lα (Ω), b ∈ Lβ (Ω) and 0 < q ≤ 1 < p. The initial data u (0) = u0 is considered in the space Lr (Ω), r ≥ 1. In the main result (0 < q < 1), we assume a, b ≥ 0 a.e. in Ω and we assume that u0 ≥ γ dΩ for some γ > 0. We find a unique solution in the space C ([0, T], Lr (Ω)) ∩ Lloc∞ ((0, T), L∞ (Ω)). © 2006 Elsevier Inc. All rights reserved.