Let $u,v,w$ be functions of $t$ defined on $[0,\infty)$, which are 3 times differentiable.
$u'''=\begin{cases}\,\,\,\,2,\,\,u<v,w\\-1,\,\,otherwise\end{cases}$
$v''',w'''$ are similarly determined.
Now consider an IVP. Given $u,v,w,u',v',w',u'',v'',w''$ at $t=0$, with $u\neq v\neq w\neq u$, can there exist $c>0$ s.t. $\forall t>c,u(t)=v(t)=w(t)$?
(Here we consider the solution of this ODE as caratheodory-solution.)