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

Abstract

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