Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains. Issue 8 (19th October 2016) - Bonforte, Matteo; Figalli, Alessio; Ros‐Oton, Xavier
Abstract: We study the positivity and regularity of solutions to the fractional porous medium equations u 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 sharp C x s / m regularity estimate up to the boundary. Our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form u t + ℒ F ( u ) = 0 in Ω, both in bounded and unbounded domains.© 2016 Wiley Periodicals, Inc.