## Analysis & PDE

• Anal. PDE
• Volume 11, Number 4 (2018), 945-982.

### 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 $∂ t u + ℒ u m = 0$, $m > 1$, where the operator $ℒ$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $Ω ⊂ ℝ N$. 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 $2 s ≤ 1 − 1 ∕ m$, 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 $ℒ u m = u$.

#### Article information

Source
Anal. PDE, Volume 11, Number 4 (2018), 945-982.

Dates
Revised: 31 July 2017
Accepted: 22 November 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.apde/1517454160

Digital Object Identifier
doi:10.2140/apde.2018.11.945

Mathematical Reviews number (MathSciNet)
MR3749373

Zentralblatt MATH identifier
06830003

#### Citation

Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis. Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains. Anal. PDE 11 (2018), no. 4, 945--982. doi:10.2140/apde.2018.11.945. https://projecteuclid.org/euclid.apde/1517454160

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