Lemma: A falling switch region above the ramp

There are $c>0$ and a nonempty interior open set $O\subset \{t>\epsilon \}$ on which ${\partial }_{t}g\le -c$.

Proof. Choose $(t_{\ast },{\mathbf{x}}_{\ast })\in U$. Then $t_{\ast }>\epsilon$ and $g(t_{\ast },{\mathbf{x}}_{\ast })=1$. Compact interior support gives a time $t_1\in (t_{\ast },T)$ with $g(t_1,{\mathbf{x}}_{\ast })=0$. The mean value theorem yields a point between them where ${\partial }_{t}g<0$. Smoothness gives the claimed open neighborhood and a strictly positive $c$.