Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains

We study the positivity and regularity of solutions to the fractional porous medium equationsu t + ( − Δ ) s u m = 0 in ( 0 , ∞ ) × Ω for m > 1 and s ∈ (0,1), with Dirichlet boundary data u = 0 in ( 0 , ∞ ) × ( ℝ N ∖ Ω ) and nonnegative initial condition u ( 0 , ⋅ ) = u 0 ≥ 0 . Our first result is a quantitative lower bound for solutions that holds for all positive times t > 0. As a consequence, we find a global Harnack principle stating that for any t > 0 solutions are comparable to d s/m , where d is the distance to ∂Ω. This is in sharp contrast with the local case s = 1, where the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ( C ∞ in x and C 1,α in t ) and establish a sharpC x s / mregularity estimate up to the boundary. Our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the formu t + ℒ F ( u ) = 0 in Ω, both in bounded and unbounded domains.© 2016 Wiley Periodicals, Inc.