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

Abstract

This paper provides sharp quantitative and constructive estimates of nonnegative solutions $u(t,x)\ge 0$ 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 $$H_0(t,u_0)\mathrm{dist}(x,\partial \Omega {)}^a\le F(u(t,x))\le H_1(t)\mathrm{dist}(x,\partial \Omega {)}^b\forall (t,x)\in (0,\infty )\times \overline{\Omega },$$ 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.

56 pages, 3 tables