Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains

Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis

doi:10.2140/apde.2018.11.945

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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

DOI: 10.2140/apde.2018.11.945

Abstract

We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_{t} u + ℒ u^{m} = 0$ , $m > 1$ , where the operator $ℒ$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $Ω \subset ℝ^{N}$ . 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, $ℒ$ can be a fractional power of a uniformly elliptic operator with $C^{1}$ coefficients. Since the nonlinearity is given by $u^{m}$ 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 $ℒ$ 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 $ℒ = {(- Δ)}^{s}$ is a spectral power of the Dirichlet Laplacian inside a smooth domain, we can prove that

when $2 s > 1 - 1 ∕ m$ , for large times all solutions behave as ${dist}^{1 ∕ m}$ near the boundary;
when $2 s \leq 1 - 1 ∕ m$ , 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 $ℒ u^{m} = u$ .

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

Vol. 11, No. 4, 2018