Calculus of Variations and Geometric Measure Theory

M. Bonforte - A. Figalli - J. L. Vazquez

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

created by figalli on 23 Nov 2017


Accepted Paper

Inserted: 23 nov 2017
Last Updated: 23 nov 2017

Journal: Anal. PDE
Year: 2017


This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + \mathcal L u^m=0$, $m>1$, where the operator $\mathcal L$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $\Omega\subset \mathbb R^N$. 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, $\mathcal L$ can be a 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 has been recently developed in \cite{BV-PPR1,BV-PPR2-1}. 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 $\mathcal L$ 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 non-matching powers for the long-time boundary behavior. More precisely, when $\mathcal L=(-\Delta)^s$ is a spectral power of the {Dirichlet} Laplacian inside a smooth domain, we can prove that:

- when $2s> 1-1/m$, for large times all solutions behave as ${\rm dist}^{1/m}$ near the boundary;

- when $2s\leq 1-1/m$, 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 $\mathcal L u^m=u$.