Strong regularizing effect of a gradient term in the heat equation with the Hardy potential

We deal with the following parabolic problem, ⎧⎪ ⎨⎪ ⎩ ut − u + |∇u|p = λ u |x|2 +f, u > 0 in Ω ×(0,T ), u(x, t) =0 on ∂Ω ×(0,T ), u(x, 0) = u0(x), x ∈ Ω, where Ω ⊂ RN, N 3, is a bounded regular domain such that 0 ∈ Ω or Ω = RN, 1< p 2, λ > 0 and f 0, u0 0 are in a suitable class of functions. Forp >p∗ ≡ N N−1 , we will show that the above problem has a solution for allλ>0, f ∈ L1(ΩT ) and u0 ∈ L1(Ω).We prove also that p∗ is optimal for the existence result. These results prove the strong regularizing effect of a gradient term in the problem studied in Baras and Goldstein (1984) [3]. The Cauchy problem is also studied. © 2009 Elsevier Inc. All rights reserved.