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

Abstract

We study a porous medium equation with fractional potential pressure: $${\partial }_{t}u=\nabla \cdot (u^{m-1}\nabla p),p=(-\Delta {)}^{-s}u,$$ for $m>1$, $0<s<1$ and $u(x,t)\ge 0$. To be specific, the problem is posed for $x\in {\mathbb{R}}^N$, $N\ge 1$, 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.

6 pages, 1 figure

References in corpus (2)

• Nonlocal porous medium equation: Barenblatt profiles and other weak solutions
• Finite difference method for a general fractional porous medium equation

Cited by in corpus (1)

• The Mesa Problem for the Fractional Porous Medium Equation