On a question of Landis and Oleinik

Abstract:

Let $P=\partial _{t}+\partial _{i}(a^{ij}\partial _{j})$ be a backward parabolic operator. It is shown that under certain conditions on $\left \{ a^{ij}\right \}$, if $u$ satisfies $\left |Pu\right |\leq$ $C(\left |u\right |+\left |\nabla u\right |)$, $\left |u(x,t)\right |\lesssim e^{C\left |x\right |^{2}}$ in $\mathbb {R}^{n}\times [0,T]$ and $\left |u(x,0)\right |\lesssim e^{-M\left |x\right |^{2}}$ for all $M>0$, then $u$ vanishes identically in $\mathbb {R}^{n}\times [0,T]$.