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Talenti inequalities are a central feature in the qualitative analysis of PDEconstrained optimal control as well as in calculus of variations. The classicalparabolic Talenti inequality states that if we consider the parabolic equation${\frac{\partial u}{\partial t}}-\Delta u=f=f(t,x)$ then, replacing, for anytime $t$, $f(t,\cdot)$ with its Schwarz rearrangement $f^\#(t,\cdot)$ increasesthe concentration of the solution in the following sense: letting $v$ be thesolution of ${\frac{\partial v}{\partial t}}-\Delta v=f^\#$ in the ball, thenthe solution $u$ is less concentrated than $v$. This property can be rephrasedin terms of the existence of a maximal element for a certain orderrelationship. It is natural to try and rearrange the source term not only inspace but also in time, and thus to investigate the existence of such a maximalelement when we rearrange the function with respect to the two variables. Inthe present paper we prove that this is not possible.

A note on the rearrangement of functions in time and on the parabolic Talenti inequality