  In this manuscript we consider a non-local porous medium equation with
non-local diffusion effects given by a fractional heat operator
\begin{equation*}
  \partial_t u = \mbox{div}(u\nabla p),\qquad
  \partial_t p = -(-\Delta)^s p + u^2, \end{equation*} in three space
dimensions for $3/4\le s < 1$ and analyze the long time asymptotics. The proof
is based on energy methods and leads to algebraic decay towards the stationary
solution $u=0$ and $\nabla p=0$ in the $L^2(\mathbb{R}^3)$-norm. The decay rate
depends on the exponent $s$. We also show weak-strong uniqueness of solutions
and continuous dependence from the initial data. As a side product of our
analysis we also show that existence of weak solutions, previously shown in
[Caffarelli, Gualdani, Zamponi 2018] for $3/4\le s \le 1$, holds for $1/2 <
s\le 1$ if we consider our problem in the torus.
