Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays. (June 2016)

• Record Type: Journal Article
• Authors: Dolbeault, Jean; Toscani, Giuseppe
• Is Part Of: Nonlinear analysis. Volume 138(2016)
• Page Start: 31
• Page End: 43
• Publication Date: 2016-06
• Subjects: primary 35K55 35K65 35B40 -- secondary 46E35 39B62 49J40; Nonlinear diffusion equations -- Gagliardo–Nirenberg–Sobolev inequalities -- Best matching Barenblatt profiles; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.11.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source-type solutions 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 evolution 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 second moment. Here the notion of best matching is defined in terms of a relative entropy.