Sharp asymptotic of solutions to some nonlocal parabolic equations

We show that if $ u $ solves the fractional parabolic equation $ (\partial_t - \Delta )^s u = Vu $ in $ B_5 \times (-25, 0] $ ($ 0<s<1 $) such that $ u(\cdot, 0) \not\equiv 0 $, then the maximal vanishing order of $ u $ in space-time at $ (0, 0) $ is upper bounded by $ C\left(1+\|V\|_{C^{1}_{(x, t)}}^{1/2s}\right) $. As $ s \to 1 $, it converges to the sharp maximal order of vanishing due to Donnelly-Fefferman and Bakri. This quantifies a space-like strong unique continuation result recently proved in [3]. The proof is achieved by means of a new quantitative Carleman estimate that we derive for the corresponding extension problem combined with a quantitative monotonicity in time result and a compactness argument.