The impact of intrinsic scaling on the rate of extinction for anisotropic non-Newtonian fast diffusion

TL;DR

This paper investigates how intrinsic scaling influences the rate at which solutions to fully anisotropic, nonlinear diffusion equations decay to extinction, using novel integral inequalities that account for anisotropic geometry.

Contribution

It introduces new integral Harnack-type inequalities for anisotropic operators, enabling analysis of decay rates without relying on comparison principles.

Findings

Different decay rates are identified depending on space geometry.

The approach develops methods for strongly nonlinear operators.

Intrinsic geometry significantly impacts extinction behavior.

Abstract

We study the decay towards the extinction that pertains to local weak solutions to fully anisotropic equations whose prototype is \[ \partial_t u= \sum_{i=1}^N \partial_i (|\partial_i u|^{p_i-2} \partial_i u), \qquad 1<p_i<2. \] Their rates of extinction are evaluated by means of several integral Harnack-type inequalities which constitute the core of our analysis and that are obtained for anisotropic operators having full quasilinear structure. Different decays are obtained when considering different space geometries. The approach is motivated by the research of new methods for strongly nonlinear operators, hence dispensing with comparison principles, while exploiting an intrinsic geometry that affects all the variables of the solution.