  In this paper we investigate existence of solutions for the system:
\begin{equation*} \left\{ \begin{array}{l} D^{\alpha}_tu=\textrm{div}(u \nabla
p),\\ D^{\alpha}_tp=-(-\Delta)^{s}p+u^{2}, \end{array} \right. \end{equation*}
in $\mathbb{T}^3$ for $0< s \leq 1$, and $0< \alpha \le 1$. The term
$D^\alpha_t u$ denotes the Caputo derivative, which models memory effects in
time. The fractional Laplacian $(-\Delta)^{s}$ represents the L\'{e}vy
diffusion. We prove global existence of nonnegative weak solutions that satisfy
a variational inequality. The proof uses several approximations steps,
including an implicit Euler time discretization. We show that the proposed
discrete Caputo derivative satisfies several important properties, including
positivity preserving, convexity and rigorous convergence towards the
continuous Caputo derivative. Most importantly, we give a strong compactness
criteria for piecewise constant functions, in the spirit of Aubin-Lions
theorem, based on bounds of the discrete Caputo derivative.
