Optimal results for the fractional heat equation involving the Hardy potential

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem (Pθ){ut+(−Δ)su=λu|x|2s+θup+cfin  Ω×(0,T),u(x,t)>0in  Ω×(0,T),u(x,t)=0in  (RN∖Ω)×[0,T),u(x,0)=u0(x)if  x∈Ω, where N>2sN>2s, 01p>1, c,λ>0c,λ>0, θ={0,1}θ={0,1}, and u0,f⩾0 are in a suitable class of functions.The main results in the article are:(1) Optimal results about existence and instantaneous and complete blow up   in the linear problem (P0)(P0), where the best constant in the fractional Hardy inequality, ΛN,sΛN,s, provides the threshold between existence and nonexistence. To obtain local sharp estimates of the solutions it is required to prove a weak Harnack inequality for a weighted operator that appears in a natural way.(2) The existence of a critical power p+(s,λ)p+(s,λ) in the semilinear problem (P1)(P1) be such that:(a) If p>p+(s,λ)p>p+(s,λ), the problem has no weak positive supersolutions and a phenomenon of complete and instantaneous blow up happens.(b) If p