Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential

Abstract In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem aligned \{ array{ll (wₜ- w) ˢ= |x|^{2s} w+wᵖ +f, & in RN (0, +), \\ w (x, t) =0, & in RN (-, 0], array. } aligned (w t - Δ w) s = λ | x | 2 s w + w p + f, in R N × (0, + ∞), w (x, t) = 0, in R N × (- ∞, 0 ], where N> 2s N > 2 s, 0 0 s 1 and 0 0 λ Λ N, s, the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent p+ (, s) p + (λ, s) and of a Fujita-type exponent F (, s) F (λ, s) such that the following holds: Let p>p_+ (, s) p > p + (λ, s). Then there are not any non-negative supersolutions. Let p p p + (λ, s). Then there exist local solutions, while concerning global solutions we need to distinguish two cases: Let 1 1 p ≤ F (λ, s). Here we show that a weighted norm of any positive solution blows up in finite time. Let F (, s) F (λ, s) p p + (λ, s). Here we prove the existence of global solutions under suitable hypotheses.