In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the reader is referred to a paper of Evans and Spruck.

Claim. If $(Q_t \mid t_0 \leq t \leq t_1)$ is a smooth mean curvature flow in $\mathbf{R}^n$, then the signed distance function $r$ to $Q_t$ satisfies the inequality \begin{equation} \tag{1} r\Big(\frac{\partial r}{\partial t} - \Delta r \Big) \geq 0 \end{equation} on the open neighborhood $U$ of $(Q_t)$ where $r$ is smooth.

Does the inequality $(1)$ really hold on the entirety of $U$?

Going through the Evans–Spruck paper, I could only make it work in a thin neighborhood of $(Q_t)$, say small enough that the spatial Hessian has $\lvert r D^2 r \rvert < 1$.

For the argument that follows in Ilmanen's book, it's perfectly fine to work with a thinner neighborhood of $(Q_t)$. What my question is getting at is whether it's necessary to update $U$ to a smaller open set in order for $(1)$ to hold.