 Solution

For each departure time $c\in [2,h]$, defineEach branch satisfies the ordinary differential equation, and both joins have matching value $-1$ and matching derivative zero. Thus $y_c$ is continuously differentiable and satisfies the same initial value problem. If $2\le c_1<c_2\le h$, the solutions differ on $(c_1,c_2)$, proving infinitely many solutions on every $[0,h]$ with $h>2$.

The uniqueness hypotheses fail when the solution reaches $-1$. IndeedNo finite Lipschitz constant works on a rectangle containing that level in its interior. The local theorem still controls the earlier branch near zero; it cannot be extended through a point where its hypotheses fail. These are waiting-time continuations at a non-Lipschitz equilibrium.