Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays

TL;DR

This paper studies functionals related to moments and entropies in nonlinear diffusion equations, revealing extremal properties of Barenblatt solutions and providing new proofs and estimates for related inequalities and solution delays.

Contribution

It introduces a method based on scaling properties to prove sharp inequalities and analyze the asymptotic behavior of solutions, including refined growth estimates and delay monotonicity.

Findings

Convexity of functionals leads to extremal properties of Barenblatt profiles.

New simple proof of sharp Gagliardo-Nirenberg-Sobolev inequalities.

Refined estimates on the growth of the second moment and solution delays.

Abstract

In this paper, we consider functionals based on moments and non-linear entropies which have a linear growth in time in case of source-type so-lutions to the fast diffusion or porous medium equations, that are also known as Barenblatt solutions. As functions of time, these functionals have convexity properties for generic solutions, so that their asymptotic slopes are extremal for Barenblatt profiles. The method relies on scaling properties of the evo-lution equations and provides a simple and direct proof of sharp Gagliardo-Nirenberg-Sobolev inequalities in scale invariant form. The method also gives refined estimates of the growth of the second moment and, as a consequence, establishes the monotonicity of the delay corresponding to the best matching Barenblatt solution compared to the Barenblatt solution with same initial sec-ond moment. Here the notion of best matching is defined in terms of a relative entropy.