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

Abstract

We provide 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 $\Omega \subset {ℝ}^N$. 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 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 $\mathcal{ℒ}$ 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 nonmatching powers for the long-time boundary behavior. More precisely, when $\mathcal{ℒ}={\left(-\Delta \right)}^s$ 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 ${dist}^{1∕m}$ 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 $\mathcal{ℒ}{u}^m=u$.