On the heat equation with drift in $L_{d+1}$

In this paper we deal with the heat equation with drift in $L_{d+1}$. Basically, we prove that, if the free term is in $L_{q}$ with high enough $q$, then the equation is uniquely solvable in a rather unusual class of functions such that $partial_{t}u, D^{2}uin L_{p}$ with $p<d+1$ and $Duin L_{q}$.