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

Abstract

This paper provides sharp quantitative and constructive estimates of nonnegative solutions $u(t,x)\geq 0$ 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.