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