Optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a Hardy potential

We deal with the following parabolic problem{ut−Δu=|∇u|p+λu|x|2+f,u>0in Ω×(0,T),u(x,t)=0on ∂Ω×(0,T),u(x,0)=u0(x),x∈Ω, where Ω⊂RNΩ⊂RN, N⩾3N⩾3, is a bounded regular domain such that 0∈Ω0∈Ω or Ω=RNΩ=RN, p>1p>1, λ⩾0λ⩾0 and f⩾0f⩾0, u0⩾0u0⩾0 are in a suitable class of functions.There are deep differences with respect to the heat equation (λ=0λ=0). The main features in the paper are the following.
• If λ>0λ>0, there exists a critical exponent p+(λ)p+(λ) such that for p⩾p+(λ)p⩾p+(λ), there is no nontrivial local solution.
• p+(λ)p+(λ) is optimal in the sense that, if p• If we consider the Cauchy problem, i.e., Ω≡RNΩ≡RN, we find the same phenomenon about the critical power p+(λ)p+(λ) as above. Moreover, there exists a Fujita type exponent  F(λ)