Non-existence and instantaneous extinction of solutions for singular nonlinear fractional diffusion equations

We show non-existence of solutions of the Cauchy problem in RN for the nonlinear parabolic equation involving fractional diffusion ∂tu+ (- Δ) sϕ(u) = 0 , with 0 < s< 1 and very singular nonlinearities ϕ. It is natural to consider nonnegative data and solutions. More precisely, we prove that when ϕ(u) = - 1 / un with n> 0 , or ϕ(u) = log u, and we take nonnegative L1 initial data, there is no solution of the problem in any dimension N≥ 2. In one space dimension the situation is not so radical, and we find the optimal range of non-existence when N= 1 in terms of s and n. As a complement, non-existence is then proved for more general nonlinearities ϕ, and it is also extended to the related elliptic problem of nonlinear nonlocal type: u+ (- Δ) sϕ(u) = f with the same type of nonlinearity ϕ