Both monotone rearrangement and relative rearrangement find their applications in evolution problems. Indeed, comparison theorems similar to those of G. Talenti have been proven by Catherine Bandle for linear equations governed by a Laplacian. She compared the regular solution of \(\displaystyle \frac {\partial u}{\partial t}-\Delta u=f\in L^2_+(Q)\) in an arbitrary cylinder \(Q=]0,T[\times \Omega \) (with Dirichlet boundary conditions) with the regular solution U of \(\displaystyle \frac {\partial U}{\partial t}-\Delta U=\underset {\widetilde \ }{f}\) in the “regular” cylinder of the same measure \(\widetilde Q=]0,T[\times \widetilde \Omega \) with comparable initial data, showing that \(\displaystyle \int _0^su_*(t,\sigma )d\sigma \leqslant \int _0^sU_*(t,\sigma )d\sigma ,\ \forall t\in [0,T[,\ \forall s\in \Omega _*\) .