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 where \(0<\alpha <1,\, \gamma >1,\, N\in \mathbb {Z}_{\geqslant 1}\) $0<\alpha <1,\gamma >1,N\in {\mathbb{Z}}_{⩾1}$ and \(\mu (x)\) $\mu (x)$ belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative \(^{C}\partial ^{\alpha }_{t}\) $_{}^C{\partial }_t^{\alpha }$ 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.