We know that, if $F:\mathbb{R}^n\to\mathbb{R}^n$ is regular enough, then a curve $c:t\to\mathbb{R}^n$ solution of the ODE $$\left\{\begin{array}{l}c'(t)=F(c(t))\\ c(0)=x\in\mathbb{R}^n\end{array}\right.$$ check the following alternative:
$c$ is defined on whole $\mathbb{R}$,
or $c$ is defined on $]t_\star,t^\star[$, with $t^\star<+\infty$, and for all compact $K\subset\mathbb{R}^n$ it exists a time $t_K\in]t_\star,t^\star[$ such that for all $t\in[t_K,t^\star[$, $c(t)\in \mathbb{R}^n\setminus K$,
or "same as 2. with $t_\star$".
My first question is: does this result have a common name in english litterature? In France we call it "explosion in finite time" or "getting out of all compact sets", but these seem not to be actual translations.
My second question is: does this alternative hold in differential manifold theory? More precisely, if I have a $\mathcal{C}^\infty$ vector field $X$ defined on a smooth manifold $M$, does a solution $c:t\to M$ of $$\left\{\begin{array}{l}c'(t)=X_{c(t)}\\ c(0)=p\in M\end{array}\right.$$ check exclusively one of the three conditions? If yes, I would be glad to have a sketch of proof/a reference. Thank you all for your attention!
I typically refer to this phenomenon as "blow up in finite time" and have heard others use the same or similar terminology. The generalization of the result you mention does hold for manifolds and you can find a proof in the second edition of Lee's introduction to smooth manifolds, chapter 9.