  We consider the Cauchy problem for fractional semilinear heat equations with
supercritical nonlinearities and establish both necessary conditions and
sufficient conditions for local-in-time solvability. We introduce the notion of
a dilation-critical singularity (DCS) of the initial data and show that such
singularities always exist for a large class of supercritical nonlinearities.
Moreover, we provide exact formulae for such singularities.
