Optimal results for the fractional heat equation involving the Hardy potential. (July 2016) - Abdellaoui, Boumediene; Medina, María; Peral, Ireneo; Primo, Ana
Abstract: In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem ( P θ ) { u t + ( − Δ ) s u = λ u | x | 2 s + θ u p + c f in  Ω × ( 0, T ), u ( x, t ) > 0 in  Ω × ( 0, T ), u ( x, t ) = 0 in  ( R N ∖ Ω ) × [ 0, T ), u ( x, 0 ) = u 0 ( x ) if  x ∈ Ω, where N > 2 s, 0 < s < 1, ( − Δ ) s is the fractional Laplacian of order 2 s, p > 1, c, λ > 0, θ = { 0, 1 }, and u 0, 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 ( P 0 ), where the best constant in the fractional Hardy inequality, Λ N, 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 ) in the semilinear problem ( P 1 ) be such that: (a) If p > p + ( s, λ ), the problem has no weak positive supersolutions and a phenomenon of complete and instantaneous blow up happens. (b) If p < p + ( s, λ ), there exists a positive solution for a suitable class of nonnegative data.