Abstract (EN)

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 solutions to the fast di usion or porous medium equations, that are also knownas Barenblatt solutions. As functions of time, these functionals have convexityproperties for generic solutions, so that their asymptotic slopes are extremalfor Barenblatt pro les. 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 givesre ned estimates of the growth of the second moment and, as a consequence,establishes the monotonicity of the delay corresponding to the best matchingBarenblatt solution compared to the Barenblatt solution with same initial second moment. Here the notion of best matching is de ned in terms of a relativeentropy.