In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made on page 60.

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 guaranteed to be smooth.

How is the inequality $(1)$ proved to hold on the entirety of $U$? Going to the paper of Evans and Spruck, as Ilmanen suggests, I could only make it work in a thin neighborhood of $(Q_t)$, say small enough that the spatial Hessian has $r \lvert D^2 r \rvert < 1$.