= Waiting-time continuation at a non-Lipschitz equilibrium
{title2=$y_c(t)=y_*-[(1-p)(t-c)]_+^{1/(1-p)},\quad0<p<1$}
= Waiting-time continuations at a non-Lipschitz equilibrium
{synonym}
For the <autonomous differential equation> $y'=-|y-y_*|^p$, the displayed family stays at $y_*$ 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 $|f(y_*+\varepsilon)-f(y_*)|/\varepsilon=\varepsilon^{p-1}$ is unbounded, so the local <Lipschitz continuity> hypothesis of the <Picard-Lindelöf theorem> fails.
A solution starting above $y_*$ reaches it in finite time. It can then use any of these continuations. For $p=1/2$, $y_*=-1$, and initial value $y(0)=0$, arrival occurs at time two; departure at $c\geq2$ 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.
\Image[/past-exam-of-the-differential-equations-1-course-of-the-university-of-oxford/2017-nonunique-continuations.png]
{title=Different <waiting-time continuations at a non-Lipschitz equilibrium> share the same earlier trajectory}
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