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

TL;DR

This paper develops sharp quantitative regularity estimates for solutions to nonlinear porous medium-type equations involving local and nonlocal operators in bounded domains, revealing new boundary behavior phenomena in the nonlocal case.

Contribution

It provides the first sharp regularity and boundary estimates for nonlocal porous medium equations, including fractional Laplacians, and uncovers novel boundary behavior phenomena.

Findings

Solutions exhibit boundary decay rates depending on the fractional order and nonlinearity.

Different solutions can show varying boundary behaviors when nonlocal operators are involved.

The results extend and sharpen existing regularity theory for degenerate parabolic equations.

Abstract

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 [14,15]. 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 $2s\le 1-1/m$, 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 ${\mathcal L}u^m=u$.