  We investigate the dissipativity properties of a class of scalar second order
parabolic partial differential equations with time-dependent coefficients. We
provide explicit condition on the drift term which ensure that the relative
entropy of one particular orbit with respect to some other one decreases to
zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic
inequality for the associated semigroup, which is derived by an adaptation of
Bakry's $\Gamma-$ calculus. As a byproduct, the systematic method for
constructing entropies which we propose here also yields the well-known
intermediate asymptotics for the heat equation in a very quick way, and without
having to rescale the original equation.
