Integral Harnack estimates and the rate of extinction of singular fractional diffusion

TL;DR

This paper establishes integral Harnack inequalities for solutions of singular fractional p-Laplacian equations and uses these to analyze the decay rate and extinction behavior of solutions over time.

Contribution

It introduces new integral Harnack estimates for singular fractional diffusion equations and applies them to determine solution decay rates without relying on an integrable time derivative.

Findings

Derived integral Harnack inequalities for fractional p-Laplacian solutions

Quantified decay rates of solutions approaching extinction time

Validated decay estimates through approximation methods

Abstract

We prove several integral Harnack-type inequalities for local weak solutions of parabolic equations with measurable and bounded coefficients, describing singular s-fractional p-Laplacian diffusion. Then we apply the aforementioned estimates to evaluate the decay rate of the local mass and supremum of the solutions as they approach a possible extinction time. Yet we show consistency of our general decay estimates by studying the extinction phenomenon for weak solutions of the Cauchy-Dirichlet problem, by means of an approximation procedure that carefully avoids the use of an integrable time derivative.