Optimal singularities of initial data of a fractional semilinear heat equation in open sets

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.