Sharp Boundary Estimates and Harnack Inequalities for Fractional Porous Medium type Equations

TL;DR

This paper establishes sharp boundary estimates and Harnack inequalities for solutions to nonlinear fractional diffusion equations with general operators and nonlinearities, revealing boundary behavior depending on initial data size.

Contribution

It provides the first comprehensive sharp boundary estimates and Harnack inequalities for fractional porous medium equations with general operators and nonlinearities, including degenerate kernels.

Findings

Derived explicit boundary Harnack inequalities.

Identified boundary behavior dependence on initial data size.

Extended regularity results for solutions.

Abstract

This paper provides sharp quantitative and constructive estimates of nonnegative solutions $u(t,x)\geq 0$ to the nonlinear fractional diffusion equation, $$\partial_t u +{\mathcal L} F(u)=0,$$ also known as filtration equation, posed in a smooth bounded domain $x\in \Omega \subset {\mathbb R}^N$ with suitable homogeneous Dirichlet boundary conditions. Both the operator ${\mathcal L}$ and the nonlinearity $F$ belong to a general class. The assumption on ${\mathcal L}$ are set in terms of the kernel of ${\mathcal L}$ and/or ${\mathcal L}^{-1}$, and allow for operators with degenerate kernel at the boundary of $\Omega$. The main examples of ${\mathcal L}$ are the three different Dirichlet Fractional Laplacians on bounded domains, and the nonlinearity can be non-homogeneous, for instance, $F(u)=u^2+u^{10}$. Previous result were known in the porous medium case, i.e. $F(u)=|u|^{m-1} u$ with $m>1$. Our aim here is to perform the next step: a delicate analysis of regularity through quantitative, constructive and sharp a priori estimates. Our main results are global Harnack type inequalities $$H_0(t,u_0)\, {\rm dist}(x, \partial \Omega)^a\leq F(u(t,x))\leq H_1(t)\, {\rm dist}(x, \partial \Omega)^b\qquad\forall (t,x)\in (0,\infty)\times \overline{\Omega},$$ where the expressions of $H_0, H_1$ and $a,b$ are explicit and may change according to ${\mathcal L}$ and $F$. The sharpness of such estimates is proven by means of examples and counterexamples: on the one hand, we can match the powers (i.e. $a=b$) when the operator has a non degenerate kernel. On the other hand, when ${\mathcal L}$ has a kernel that degenerates at the boundary $\partial\Omega$, there appear an intriguing anomalous boundary behaviour: the size of the initial data determines the sharp boundary behaviour of the solution, different for ``small'' and ``large'' initial data. We conclude the paper with higher regularity results.