Nonlinear Differ. Equ. Appl. (2018) 25:30 c  2018 Springer International Publishing AG, part of Springer Nature 1021-9722/18/040001-34 published online June 12, 2018 https://doi.org/10.1007/s00030-018-0521-y Nonlinear Differential Equations and Applications NoDEA On a fractional quasilinear parabolic problem: the influence of the Hardy potential Boumediene Abdellaoui, Ahmed Attar, Rachid Bentifour and Ireneo Peral Abstract. The aim goal of this paper is to treat the following problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u t +(-Δ) s p u = λ u p-1 |x| ps in ΩT =Ω × (0,T ), u ≥ 0 in IR N × (0,T ), u =0 in (IR N \ Ω) × (0,T ), u(x, 0) = u0(x) in Ω, where Ω is a bounded domain containing the origin, (-Δ) s p u(x, t) := P.V  IR N |u(x, t) - u(y,t)| p-2 (u(x, t) - u(y,t)) |x - y| N+ps dy with 1 <p< N s ,s ∈ (0, 1) and f,u0 are non negative functions. The main goal of this work is to discuss the existence and non existence of solutions according to the values of p and λ. Mathematics Subject Classification. 35K59, 35K65, 35K67, 35K92, 35B09. Keywords. Nonlinear nonlocal parabolic problems, Hardy potential, Caffarelli– Kohn–Nirenberg inequalities, Degenerate weights, Finite time extinction, Non existence result. 1. Introduction In this paper will be studied the following parabolic problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u t +(−Δ) s p u = λ u p−1 |x| ps in Ω T =Ω × (0,T ), u ≥ 0 in IR N , u =0 in (IR N \ Ω) × (0,T ), u(x, 0) = u 0 (x) in Ω, (1.1) This work is supported by Ministerio de Economia y Competitividad under Grants MTM2013-40846-P and MTM2016-80474-P (Spain).