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 { C partial derivative(alpha)(t) u = Delta u + |u|(gamma-1)u, x is an element of R-N , t is an element of]0, T [, (P) {u ( x , 0 ) = mu(x), x is an element of R-N , where 0 < alpha < 1, gamma > 1, N is an element of Z 1 and mu(x) belongs to inhomogeneous/homogeneous Besov-Morrey spaces. The fractional derivative C partial derivative(alpha)(t)u is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.