  We study a porous medium equation with fractional potential pressure: $$
\partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-\Delta)^{-s}u, $$ for
$m>1$, $0<s<1$ and $u(x,t)\ge 0$. To be specific, the problem is posed for
$x\in \mathbb{R}^N$, $N\geq 1$, and $t>0$. The initial data $u(x,0)$ is assumed
to be a bounded function with compact support or fast decay at infinity. We
establish existence of a class of weak solutions for which we determine
whether, depending on the parameter $m$, the property of compact support is
conserved in time or not, starting from the result of finite propagation known
for $m=2$. We find that when $m\in [1,2)$ the problem has infinite speed of
propagation, while for $m\in [2,\infty)$ it has finite speed of propagation.
Comparison is made with other nonlinear diffusion models where the results are
widely different.
