  The goal of this paper is to give a non-local sufficient condition for
generalized Poincar\'e inequalities, which extends the well-known Bakry-Emery
condition. Such generalized Poincar\'e inequalities have been introduced by W.
Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow,
the exponential decay of some generalized entropies which interpolate between
the $L^2$ norm and the usual entropy. Our criterion improves on results which,
for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock
type perturbation results. In a second step, we apply the same strategy to
non-linear equations of porous media type. This provides new interpolation
inequalities and decay estimates for the solutions of the evolution problem.
The criterion is again a non-local condition based on the positivity of the
lowest eigenvalue of a Schr\"odinger operator. In both cases, we relate the
Fisher information with its time derivative. Since the resulting criterion is
non-local, it is better adapted to potentials with, for instance, a
non-quadratic growth at infinity, or to unbounded perturbations of the
potential.
