Existence and stabilization results for a singular parabolic equation involving the fractional laplacian

Abstract

In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity (Pts) {ut + (−∆)su = u−q + f(x, u), u > 0 in (0, T) × Ω, u = 0 in (0, T) × (Rn \ Ω), u(0, x) = u0(x) in Rn, where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n > 2s, s ∈ (0, 1), q > 0, q(2s − 1) < (2s + 1), u0 ∈ L∞(Ω) ∩ X0(Ω) and T > 0. We suppose that the map (x, y) ∈ Ω × R+ → f(x, y) is a bounded from below Carathéodary function, locally Lipschitz with respect to the second variable and uniformly for x ∈ Ω and it satisfies lim supf(x, y) < λs1(Ω), y→+∞ y (0.1) where λs1(Ω) is the first eigenvalue of (−∆)s in Ω with homogeneous Dirichlet boundary condition in Rn\Ω. We prove the existence and uniqueness of a weak solution to (Pts) on assuming u0 satisfies an appropriate cone condition. We use the semi-discretization in time with implicit Euler method and study the stationary problem to prove our results. We also show additional regularity on the solution of (Pts) when we regularize our initial function u0.