Finite and infinite speed of propagation for porous medium equations with fractional 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. To be specific, 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 the 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∈[1,2) the problem has infinite speed of propagation, while for m∈[2,∞) it has finite speed of propagation. Comparison is made with other nonlinear diffusion models where the results are widely different.