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 $p \lt d + 1$ and $D_u \in L_q$.

Keywords

heat equation, singular first-order terms, non-perturbative technique

2010 Mathematics Subject Classification

35B45, 35B65

Full Text (PDF format)

Received 16 March 2021

Accepted 7 January 2022

Published 1 March 2023