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 Γ-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.