On the forward in time propagation of zeros in fractional heat type problems

In this short note we prove that if $u$ solves $({\mathrm{\partial }}_t-\mathrm{\Delta }{)}^{s}u=Vu$ in ${\mathbb{R}}_x^n\times {\mathbb{R}}_t$, and vanishes to infinite order at a point $(x_0,t_0)$, then $u\equiv 0$ in ${\mathbb{R}}_x^n\times {\mathbb{R}}_t$. This sharpens (and completes) our earlier result that proves $u(\cdot ,t)\equiv 0$ for $t\le t_0$ if it vanishes to infinite order at $(x_0,t_0)$.