Vol. 11, No. 4, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains

Matteo Bonforte, Alessio Figalli and Juan Luis Vázquez

Vol. 11 (2018), No. 4, 945–982
Abstract

We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form tu + um = 0, m > 1, where the operator belongs to a general class of linear operators, and the equation is posed in a bounded domain Ω N . As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, can be a fractional power of a uniformly elliptic operator with C1 coefficients. Since the nonlinearity is given by um with m > 1, the equation is degenerate parabolic.

The basic well-posedness theory for this class of equations was recently developed by Bonforte and Vázquez (2015, 2016). Here we address the regularity theory: decay and positivity, boundary behavior, Harnack inequalities, interior and boundary regularity, and asymptotic behavior. All this is done in a quantitative way, based on sharp a priori estimates. Although our focus is on the fractional models, our results cover also the local case when is a uniformly elliptic operator, and provide new estimates even in this setting.

A surprising aspect discovered in this paper is the possible presence of nonmatching powers for the long-time boundary behavior. More precisely, when = (Δ)s is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that

  • when 2s > 1 1m, for large times all solutions behave as dist1m near the boundary;
  • when 2s 1 1m, different solutions may exhibit different boundary behavior.

This unexpected phenomenon is a completely new feature of the nonlocal nonlinear structure of this model, and it is not present in the semilinear elliptic equation um = u.

Keywords
nonlocal diffusion, nonlinear equations, bounded domains, a priori estimates, positivity, boundary behavior, regularity, Harnack inequalities
Mathematical Subject Classification 2010
Primary: 35B45, 35B65, 35K55, 35K65
Milestones
Received: 2 February 2017
Revised: 31 July 2017
Accepted: 22 November 2017
Published: 12 January 2018
Authors
Matteo Bonforte
Departamento de Matemáticas
Universidad Autónoma de Madrid
Campus de Cantoblanco
Madrid
Spain
Alessio Figalli
ETH Zürich
Department of Mathematics
Zürich
Switzerland
Juan Luis Vázquez
Departamento de Matemáticas
Universidad Autónoma de Madrid
Campus de Cantoblanco
Madrid
Spain