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

We deal with the following parabolic problem,{ut−Δu+|∇u|p=λu|x|2+f,u>0inΩ×(0,T),u(x,t)=0on∂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, 10 and f⩾0, u0⩾0 are in a suitable class of functions. For p>p∗≡NN−1, we will show that the above problem has a solution for all λ>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.