Optimal results for the fractional heat equation involving the Hardy potential. (July 2016)

• Record Type: Journal Article
• Authors: Abdellaoui, Boumediene; Medina, María; Peral, Ireneo; Primo, Ana
• Is Part Of: Nonlinear analysis. Volume 140(2016)
• Page Start: 166
• Page End: 207
• Publication Date: 2016-07
• Subjects: 35B25 -- 35K58 -- 35B33 -- 47G20; Singular fractional Laplacian heat equation -- Hardy's inequality -- Existence and nonexistence results -- Instantaneous and complete blow up -- Harnack inequality; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.03.013
• Languages: English
• ISSNs: 0362-546X

Abstract

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.