Waiting-time continuation at a non-Lipschitz equilibrium ($y_c(t)=y_{\ast }-[(1-p)(t-c){]}_{+}^{1/(1-p)},0<p<1$)

For the autonomous differential equation $y^{\prime }=-|y-y_{\ast }{|}^p$, the displayed family stays at $y_{\ast }$ until time $c$ and then decreases below it. Here $[z{]}_{+}=\max (z,0)$. Direct differentiation verifies the equation on each side; since $1/(1-p)>1$, the derivative tends to zero at departure, making the joined function continuously differentiable. Distinct departure times give different solutions from the same equilibrium initial condition. The ratio $|f(y_{\ast }+\epsilon )-f(y_{\ast })|/\epsilon ={\epsilon }^{p-1}$ is unbounded, so the local Lipschitz continuity hypothesis of the Picard-Lindelöf theorem fails.

A solution starting above $y_{\ast }$ reaches it in finite time. It can then use any of these continuations. For $p=1/2$, $y_{\ast }=-1$, and initial value $y(0)=0$, arrival occurs at time two; departure at $c\ge 2$ gives $y=-1-(t-c{)}^2/4$ afterwards. Waiting-time solutions of a sublinear Bernoulli equation give a related mechanism for departures in the opposite direction.