  This paper is devoted to the computation of the asymptotic boundary terms in
entropy methods applied to a fast diffusion equation with weights associated
with Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, only
elliptic equations have been considered and our goal is to justify, at least
partially, an extension of the carr{\'e} du champ / Bakry-Emery / R{\'e}nyi
entropy methods to parabolic equations. This makes sense because evolution
equations are at the core of the heuristics of the method even when only
elliptic equations are considered, but this also raises difficult questions on
the regularity and on the growth of the solutions in presence of weights.We
also investigate the relations between the optimal constant in the entropy -
entropy production inequality, the optimal constant in the information -
information production inequality, the asymptotic growth rate of generalized
R{\'e}nyi entropy powers under the action of the evolution equation and the
optimal range of parameters for symmetry breaking issues in
Caffarelli-Kohn-Nirenberg inequalities, under the assumption that the weights
do not introduce singular boundary terms at x=0. These considerations are new
even in the case without weights. For instance, we establish the equivalence of
carr{\'e} du champ and R{\'e}nyi entropy methods and explain why entropy
methods produce optimal constants in entropy - entropy production and
Gagliardo-Nirenberg inequalities in absence of weights, or optimal symmetry
ranges when weights are present.
