@article{pre05312043,
author="Dolbeault, J. and Nazaret, B. and Savar\'e, G.",
title="{On the Bakry-Emery criterion for linear diffusions and weighted porous
media equations.}",
language="English",
journal="Commun. Math. Sci. ",
volume="6",
number="2",
pages="477-494",
year="2008",
abstract="{Summary: 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.}",
keywords="{parabolic equations; diffusion; Ornstein-Uhlenbeck operator; porous
media; Poincar\'e inequality; logarithmic Sobolev inequality; convex Sobolev
inequality; interpolation; decay rate; entropy; free energy; Fisher
information}",
classmath="{*35B40 (Asymptotic behavior of solutions of PDE)
35K55 (Nonlinear parabolic equations)
39B62 (Systems of functional equations)
35J10 (Schroedinger operator)
35K20 (Second order parabolic equations, boundary value problems)
35K65 (Parabolic equations of degenerate type)
}",
}