Non-uniqueness for a critical non-linear heat equation

Here we consider a class of non-linear heat equation with polynomial non-linearity. We prove a non-uniqueness result for mild solutions which take values in a critical Lebesgue space. To this end we extend to the entire space a counter-example of Ni and Sacks in the case where the underlying space is the ball of center 0 and of radius 1. We also propose a new criterion of uniqueness optimal with respect to the given counter-examples. The proof of our results lie on some estimates for the heat kernel in Lorentz spaces introduced by Meyer in the Navier-Stokes context.