The Cauchy-Dirichlet Problem for Singular Nonlocal Diffusions on Bounded Domains

TL;DR

This paper develops a comprehensive theory for singular nonlocal diffusion equations on bounded domains, establishing existence, uniqueness, smoothing effects, boundary behavior, and extinction properties with explicit estimates.

Contribution

It introduces new weighted smoothing estimates and compares methods for proving effects, advancing understanding of singular nonlocal diffusions like the fractional fast diffusion equation.

Findings

Existence and uniqueness for broad data classes

New weighted smoothing estimates for solutions

Explicit bounds on extinction time and rates

Abstract

We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with smooth boundary and $N\ge 1$. The linear diffusion operator $\mathcal{L}$ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $u^m=|u|^{m-1}u$ with $0<m<1$. The prototype equation is the Fractional Fast Diffusion Equation (FFDE), when $\mathcal{L}$ is one of the three possible Dirichlet Fractional Laplacians on $\Omega$. Our main results shall provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far, both for nonnegative and signed solutions; sharp smoothing estimates: besides the classical $L^p-L^\infty$ smoothing effects, we provide new weighted estimates, which represent a novelty also in well studied local case, i.e. for solutions to the FDE $u_t=\Delta u^m$. We compare two strategies to prove smoothing effects: Moser iteration VS Green function method. Due to the singular nonlinearity and to presence of nonlocal diffusion operators, the question of how solutions satisfy the lateral boundary conditions is delicate. We answer with quantitative upper boundary estimates that show how boundary data are taken. Once solutions exists and are bounded we show that they extinguish in finite time and we provide upper and lower estimates for the extinction time, together with explicit sharp extinction rates in different norms. The methods of this paper are constructive, in the sense that all the relevant constants involved in the estimates are computable.