  We consider necessary conditions and sufficient conditions on the solvability
of the Cauchy--Dirichlet problem for a fractional semilinear heat equation in
open sets (possibly unbounded and disconnected) with a smooth boundary. Our
conditions enable us to identify the optimal strength of the admissible
singularity of initial data for the local-in-time solvability and they differ
in the interior of the set and on the boundary of the set.
