I am dealing with the following problem at the moment: Show that the differential equation \begin{align} \dot u=\sqrt{|u|} \end{align} does not define a flow on $\mathbb{R}$. Are there nontrivial intervals $I$ and $D$ such that the equation defines a local flow on $I\times D$? If so, give an example, otherwise explain why such $I$ and $D$ cannot exist.
I am really new into the theory of dynamical systems and flows, but I think for the first part:
The ODE doesn't define a flow on $\mathbb{R}$ because it has no unique solution on $\mathbb{R}$. I mean there are at lest two different solutions: the constant solution $u(t)=0\,\, \forall t\in \mathbb{R}$ and $u(t)=\dfrac{1}{4}t^2$ for $t>0$ and $u(t)=0$ for $t\leq 0$.
This is because $f(x)=\sqrt{|x|}$ is not locally Lipschitz on $\mathbb{R}$. In fact its derivate explodes for $t\rightarrow 0$.
Now to the second part:
I am not really sure what to do here. I mean if such $I$ and $D$ exist $D$ should not contain $0$ because then we always get the constant solution $0$ and we haven't got uniqueness. But I can't really find a solution for which $u(0)\not = 0$. Nor can I proof that such $I$ and $D$ dont exist.
I would appreciate it very much if someone could help me with that one.