We show how to extend the monotonicity approach of (Guterres et al. J. Differ. Equ. 356:407–431, 2023) to parabolic equations of arbitrary order and illustrate its application to the derivation of upper and lower \(L^{2}\) estimates for solution derivatives of all orders, when they exist, once such estimates have been established for the solutions themselves. This method was originally developed at the Universidade Federal do Rio Grande do Sul in 2020 by P. Zingano and collaborators to investigate the decay of higher order derivatives of solutions to a large class of viscous dissipative systems, with many applications. Upper and lower estimates for solutions and their derivatives are very important in the investigation of a system’s large time behavior. In our extension of the method to parabolic equations of any order, we also indicate a few shortcuts and improvements to the original method that can be made at some points in the analysis.
