\begin{equation} \begin{cases} a(x, t)u_t+(-\Delta)^{\sigma}u+b(x,t)u=f(x,t), & \text{in } \mathbb{R}^n \times [0, T) \\ u(x,0)=u_0(x), & \text{in } \mathbb{R}^n \end{cases} \end{equation}
I'm wondering how to prove the existence of solution of the above parabolic fractional equation and also the Schauder estimates (Regularity of Hölder spaces). I know we can use the freezing coeffients method which means we just consider the fraction heat equation $u_t+(-\Delta)^{\sigma}u=f$. I still don't know the existence and Schauder estimates of this simpler equation. Finally, I would like to know how to prove the existence of unique smooth solution of the linear parabolic fractional equation if we know all the coeffients, $f(x, t)$ and initial value $u_0(x)$ are smooth functions.
