Upper and lower $\dot{H}^{m}$ estimates for solutions to parabolic equations

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.