The Cauchy-Dirichlet Problem for the Fast Diffusion Equation on Bounded Domains

TL;DR

This paper surveys recent mathematical results on the fast diffusion equation with Dirichlet boundary conditions, highlighting existence, regularity, and asymptotic behavior, and introduces new global Harnack estimates and asymptotic results in the subcritical regime.

Contribution

It provides a comprehensive survey of recent advances and introduces new global Harnack estimates and asymptotic results for the fast diffusion equation on bounded domains.

Findings

New global Harnack estimates in the subcritical regime

Enhanced understanding of asymptotic behavior of solutions

Summary of existence, regularity, and positivity results

Abstract

The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean domains. In addition to its physical relevance, there are many aspects that make this equation particularly interesting from the pure mathematical perspective. For instance: mass is lost and solutions may extinguish in finite time, merely integrable data can produce unbounded solutions, classical forms of Harnack inequalities (and other regularity estimates) fail to be true, etc. In this paper, we first provide a survey (enriched with an extensive bibliography) focussing on the more recent results about existence, uniqueness, boundedness and positivity (i.e., Harnack inequalities, both local and global), and higher regularity estimates (also up to the boundary and possibly up to the extinction time). We then prove new global (in space and time) Harnack estimates in the subcritical regime. In the last section, we devote a special attention to the asymptotic behaviour, from the first pioneering results to the latest sharp results, and we present some new asymptotic results in the subcritical case.