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arXiv · 1311.7007

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

Diana Stan·Félix del Teso·Juan Luis Vázquez
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Abstract

We study a porous medium equation with fractional potential pressure: $$ \partial_t u= \nabla \cdot (u^{m-1} \nabla p), \quad p=(-Δ)^{-s}u, $$ for $m>1$, $0 0$. 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.

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Diana Stan, Félix del Teso, Juan Luis Vázquez. 2013-11-27. Finite and infinite speed of propagation for porous medium equations with fractional pressure. https://arxiv.org/abs/1311.7007

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