Functional inequalities and applications to doubly nonlinear diffusion equations

TL;DR

This paper establishes weighted Hardy and Hardy-Poincaré inequalities with optimal constants and applies them to analyze the convergence rates of solutions to doubly nonlinear fast diffusion equations towards their Barenblatt profiles.

Contribution

It provides necessary and sufficient conditions for weighted inequalities and demonstrates their application in quantifying solution convergence in nonlinear diffusion equations.

Findings

Derived optimal weighted inequalities with explicit constants.

Identified conditions for the inequalities to hold.

Quantified convergence rates of solutions to the Barenblatt profile.

Abstract

We study weighted inequalities of Hardy and Hardy-Poincar\'e type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.