Decay estimates in time for fractional evolution equations

Abstract

We consider the homogeneous time-fractional diffusion equation

Here, ${\partial }^{\alpha }$ denotes the Riemann–Liouville fractional derivative of order $\alpha \in (0,1)$ with respect to time and $A$ is a uniformly elliptic operator on ${\mathbb{R}}^d$. We prove decay estimates in time and other regularity properties for the solution of the above equation.