Finite and infinite speed of propagation for porous medium equations with nonlocal pressure

We study a porous medium equation with fractional potential pressure:∂tu=∇⋅(um−1∇p),p=(−Δ)−su, for m>1m>1, 00t>0. The initial data u(x,0)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 the property of compact support is conserved in time depending on the parameter m  , starting from the result of finite propagation known for m=2m=2. We find that when m∈[1,2)m∈[1,2) the problem has infinite speed of propagation, while for m∈[2,3)m∈[2,3) it has finite speed of propagation. In other words m=2m=2 is critical exponent regarding propagation. The main results have been announced in the note [29].