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 $\partial_{t}u, D^{2}u\in L_{p}$ with $p<d+1$ and $Du\in L_{q}$.

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