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We provide a scheme of a recent stability result for a family ofGagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to animproved entropy - entropy production inequality associated with an appropriatefast diffusion equation (FDE) written in self-similar variables. This resultcan be rephrased as an improved decay rate of the entropy of the solution of(FDE) for well prepared initial data. There is a family ofCaffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similarstructure. When the exponents are in a range for which the optimal functionsfor (CKN) are radially symmetric, we investigate how the methods for (GNS) canbe extended to (CKN). In particular, we prove that the solutions of theevolution equation associated to (CKN) also satisfy an improved decay rate ofthe entropy, after an explicit delay. However, the improved rate is obtainedwithout assuming that initial data are well prepared, which is a majordifference with the (GNS) case.

Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations