Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation:


‎$‎‎$‎
‎\begin{cases}‎
‎u\_t = (-\Delta)^s u + ‎f(x,t) & \quad \mathrm{in} ‎\Omega \times (0,T),\\ u(x,0)=u\_0 & \quad \mathrm{in} ‎\mathbb{R}^N‎, ‎\\ u(x,t)=0 & \quad \mathrm{in} ‎(\mathbb{R}^N \setminus \Omega )\times (0,T)‎.
‎\end{cases}‎
‎$‎‎$‎
I want to know that, does $(-\Delta)^s$ generate a semigroup and is it analytic? 
Is the regularity results for this problem, depending on the regularities of $f(x,t)$ and $u\_0$, well-known?


Can someone give a survey of reference for these questions?



Question about the regularity of fractional Heat equation
It depends on $s$, if $1/2\leq s \leq 1$, then you get analyticity. If $0<s<1/2$, then you get Gevrey class only. See Section 8.3, arXiv:1606.00873


