On a question of Landis and Oleinik

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 $\left\vert Pu\right\vert\leq$ $C(\left\vert u\right\vert+\left\vert\nabla u\right\vert)$, $\left\vert u(x,t)\right\vert\lesssim e^{C\left\vert x\right\vert^{2}}$ in $\mathbb{R}^{n}\times[0,T]$ and $\left\vert u(x,0)\right\vert\lesssim e^{-M\left\vert x\right\vert^{2}}$ for all $M>0$, then $u$ vanishes identically in $\mathbb{R}^{n}\times[0,T]$.