Let $(M,g(t))$, $t \in [0,T]$, be a smooth Ricci flow on a compact manifold. Let $u: M \times [0,T] \to \mathbb R$ be smooth and suppose
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\begin{align*}
\partial_t u \ge \Delta_{g(t)} u + F(u,t)
\end{align*}
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for a [continuous function](/page/Continuous%20Function) $F: \mathbb R \times [0,T] \to \mathbb R$ that is locally Lipschitz in the first variable uniformly on compact subsets of $\mathbb R \times [0,T]$. If $\varphi: I \to \mathbb R$, where $I \subset [0,T]$ is an interval containing $0$, solves