Sharp Boundary Estimates and Harnack Inequalities for Fractional Porous Medium type Equations
Matteo Bonforte, Carlos Fuertes-Moran
TL;DR
The paper studies sharp boundary estimates and Harnack inequalities for nonnegative solutions of $\partial_t u + \mathcal{L} F(u)=0$ in bounded domains with Dirichlet boundary conditions, allowing a broad class of nonlocal operators $\mathcal{L}$ and nonlinearities $F$. It develops a constructive framework based on Green-function estimates, weighted $L^1_{\Phi_1}$-smoothing, and barrier methods to prove global Harnack principles of the form $H_0(t,u_0)\,\Phi_1(x)^a \le F(u(t,x)) \le H_1(t)\,\Phi_1(x)^b$, with explicit exponents determined by the operator and nonlinearity; it also uncovers anomalous boundary behavior when the Green kernel degenerates at the boundary, depending on initial data size and time regime. The authors establish absolute and smoothing upper bounds, weighted $L^1$-estimates, and lower bounds that imply infinite propagation speed in the nonlocal setting, along with interior Hölder regularity under suitable kernel assumptions. The results cover classical and fractional Dirichlet Laplacians (RFL, CFL, SFL) and a broad class of kernels, and they are supported by a detailed sharpness analysis, examples, and a comprehensive plan that includes tables of the main estimates. These findings have implications for numerical methods and the qualitative understanding of boundary behavior in nonlinear nonlocal diffusion problems.
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 $x\in Ω\subset {\mathbb R}^N$ 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 $Ω$. 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 $$H_0(t,u_0)\, {\rm dist}(x, \partial Ω)^a\leq F(u(t,x))\leq H_1(t)\, {\rm dist}(x, \partial Ω)^b\qquad\forall (t,x)\in (0,\infty)\times \overlineΩ,$$ 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 $\partialΩ$, 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.