Existence of solutions for time fractional semilinear parabolic equations in Besov--Morrey spaces

We consider the Cauchy problem for a time fractional semilinear heat equation with initial data belonging to inhomogeneous/homogeneous Besov–Morrey spaces. We present sufficient conditions for the existence of local/global-in-time solutions to the Cauchy problem, which cover all existing results in the literature and can be applied to a large class of initial data.