Sharp asymptotic of solutions to some nonlocal parabolic equations

We show that if $u$ solves the fractional parabolic equation $({\mathrm{\partial }}_t-\mathrm{\Delta }{)}^{s}u=Vu$ in $B_5\times (-25,0]$ ($0<s<1$) such that $u(\cdot ,0)\equiv ̸0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is upper bounded by $C(1+\mathrm{\parallel }V{\mathrm{\parallel }}_{C_{(x,t)}^1}^{1\mathrm{/}2s})$. 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.