  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.
