  We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$
for the nonlinear parabolic equation involving fractional diffusion $\partial_t
u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities
$\phi$ . More precisely, we prove that when $\phi(u)=-1/u^n$ with $n>0$, or
$\phi(u) = \log u$, and we take nonnegative $L^1$ initial data, there is no
(nonnegative) solution of the problem in any dimension $N\ge 2$. 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 $\phi$, and it is also extended to the related elliptic problem
of nonlinear nonlocal type: $u + (-\Delta)^s \phi(u) = f$ with the same type of
nonlinearity $\phi$.
