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Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains
Author(s) -
Bonforte Matteo,
Figalli Alessio,
RosOton Xavier
Publication year - 2017
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21673
Subject(s) - mathematics , bounded function , boundary (topology) , porous medium , dirichlet distribution , mathematical analysis , dirichlet boundary condition , boundary value problem , pure mathematics , porosity , geotechnical engineering , engineering
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.