Let $(M,g(t))$, $t \in [0,T]$, be a smooth Ricci flow on a connected compact manifold. Let $u:M\times[0,T]\to\mathbb{R}$ be a smooth function with $u\geq 0$, and let $a:M\times[0,T]\to\mathbb{R}$ be a bounded smooth function. Suppose
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\begin{align*}
\partial_t u \ge \Delta_{g(t)}u + a(x,t)u
\end{align*}
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on $M\times(0,T]$. If $u(\cdot,0) \ge 0$ and there exist $x_0\in M$ and $t_0\in(0,T]$ such that $u(x_0,t_0)=0$, then $u(\cdot,t)=0$ for all $t \in [0,t_0]$.