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

We show non-existence of solutions of the Cauchy problem in R for the nonlinear parabolic equation involving fractional diffusion ∂tu + (-∆) s φ(u) = 0, with 0 < s < 1 and very singular nonlinearities φ . More precisely, we prove that when φ(u) = −1/u with n > 0, or φ(u) = log u, and we take nonnegative L initial data, there is no (nonnegative) solution of the problem in any dimension N ≥ 2. We find the range of non-existence when N = 1 in terms of s and n. The range of exponents that we find for non-existence both for parabolic and elliptic equations are optimal. 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 + (-∆) s φ(u) = f with the same type of nonlinearity φ.