  In this article we prove results concerning upper and lower decay estimates
for homogeneous Sobolev norms of solutions to a rather general family of
parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and
Zingano, we use eventual regularity of solutions to directly work with smooth
solutions in physical space, bootstrapping decay estimates from the $L^2$ norm
to higher order derivatives. Besides obtaining upper and lower bounds through
this method, we also obtain reverse results: from higher order derivatives
decay estimates, we deduce bounds for the $L^2$ norm. We use these general
results to prove new decay estimates for some equations and to recover some
well known results.
