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

Abstract

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 $Du\in L_q$.

14 pages, more general situation added