I'm interested in the following Cauchy problem for a linear diffusion equation
$$
\begin{cases}
{^C}\!D^{a}\_tu(t,x) = \sigma\Delta u(t,x),\\
u(0)=u\_0\in X.
\end{cases}
$$


where ${^C}\!D^{\sigma}\_t$
denotes the Caputo fractional derivative of order $a\in (0,1)$, and $X$ is a Banach space.  
 I wonder if there are any well-posedness results and characterizations of the $C\_0$ semigroup and its properties. Any reference would be great. Thanks.



Fractional reaction-diffusion with Caputo derivative
In $L^2(\Omega)$, see



> 
> K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl.,382 (2011), 426–447.
> 
> 
> 


For a general theory in Banach spaces, see



> 
> E. G. Bajlekova, Fractional evolution equations in Banach spaces, Thesis, Eindhoven, 2001.
> 
> 
> 


Note that solutions to fractional evolution equations do not satisfy the semigroup property!


