Pardon me if the question is trivial, but I am failing to decide it.
Assume that we are given an ODE system $\dot{x} = f(x)$ with positive initial conditions $x(0)$ and know that $f$ is locally Lipschitz on $\mathbb{R}^n_{>0}$ (In particular, $f$ is defined everywhere on $\mathbb{R}^n_{>0}$). Moreover, it is known that any solution to the above ODE system will remain in $\mathbb{R}^n_{>0}$.
Is it then true that the unique noncontinuable solution $x(t)$ is defined at least on all $t \geq 0$? A counterexample to this statement would need to diverge in norm on a finite time interval. Unfortunately, I do not see whether this is possible ...
Would be great if someone could help me out.
Cheers, Max
Hans Lundmark gave the counterexample $\dot x = x^2$: indeed, the solution with positive initial value $x(0)=x_0$ is $$x(t) = \frac{x_0}{1-x_0 t}$$ which flows up at time $t=1/x_0$.
The global Lipschitz condition would suffice for global existence. There are other situations when one can prove global existence, by inspecting the behavior of solutions in more details (sometimes they happen to stay in the region where $f$ is Lipschitz).