Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary

Boumediene Abdellaoui (1) , Kheireddine Biroud (1) , El-Haj Laamri (2)

1 Université Aboubekr Belkaid - University of Belkaïd Abou Bekr [Tlemcen]
2 IECL - Institut Élie Cartan de Lorraine

In this work, we consider a nonlocal semilinear parabolic problem related to a fractional Hardy inequality with singular weight at the boundary. More precisely, we consider the problem (P)      ut + (−∆) s u = λ u p d 2s in Ω T = Ω × (0, T), u(x, 0) = u 0 (x) in Ω, u = 0 in (IR N \ Ω) × (0, T), where 0 < s < 1, Ω ⊂ IR N is a bounded regular domain, d(x) = d(x, ∂Ω), p > 0 and λ > 0 is a positive constant. The initial data u 0 0 is a nonnegative function in a suitable Lebesgue space that we make precise later. The main goal of this work is to analyze the interaction between the parameters s, p and λ in order to show the existence or the nonexistence of solution to problem (P) in a suitable sense. We will show that our results have a significative difference with respect to the local case s = 1.

Domaines: Equations aux dérivées partielles [math.AP]

Boumediene Abdellaoui, Kheireddine Biroud, El-Haj Laamri. Existence and nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary. 2018. ⟨hal-01848861⟩