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

被引:29
|
作者
Bonforte, Matteo [1 ]
Figalli, Alessio [2 ]
Ros-Oton, Xavier [2 ]
机构
[1] Univ Autonoma Madrid, Dept Matemat, Campus Cantoblanco, E-28049 Madrid, Spain
[2] Univ Texas Austin, Dept Math, 2515 Speedway Stop C1200, Austin, TX 78712 USA
基金
美国国家科学基金会;
关键词
DEGENERATE DIFFUSION-EQUATIONS; FREE-BOUNDARY; ANOMALOUS DIFFUSION; NONLINEAR DIFFUSION; ASYMPTOTIC-BEHAVIOR; GAS-FLOW; CONTINUITY; LEVY; INTERFACES; UNIQUENESS;
D O I
10.1002/cpa.21673
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We study the positivity and regularity of solutions to the fractional porous medium equations ut+(-)sum=0 in (0,)x for m > 1 and s (0,1), with Dirichlet boundary data u = 0 in (0,)x(N\) and nonnegative initial condition u(0,)=u00. 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 . 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,C- in t) and establish a sharp Cxs/m</mml:msubsup> 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 <mml:msub>ut+LF(u)=0 in , both in bounded and unbounded domains.(c) 2016 Wiley Periodicals, Inc.
引用
收藏
页码:1472 / 1508
页数:37
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