Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

TL;DR

This paper establishes the optimal decay and convergence rates for solutions to the nonlinear fast diffusion equation by identifying optimal constants in Hardy-Poincaré inequalities, advancing the understanding of nonlinear diffusion dynamics.

Contribution

It provides the first precise decay rates and convergence results for solutions in self-similar variables, connecting functional inequalities with nonlinear diffusion behavior.

Findings

Optimal decay rates for solutions are identified.

Convergence to Barenblatt profiles is proven with optimal rates.

The work links Hardy-Poincaré inequalities to diffusion dynamics.

Abstract

The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincar\'e inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations.