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

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