Does a bound on a solution to an ODE allow for it to be defined over all $t \in \Bbb R$? Consider the ODE $$ x^{(n)}(t) = f(t, x, x^{(1)}, \dots, x^{(n-1)})$$
Much of the books I have read through talk about results for very loose conditions on $f$.  My first question is are there any references which look strictly at the case for $f$ smooth in all its variables?  I am finding it hard to filter through the results for nice $f$ and those for $f$ which can be a lot nastier and not of importance to me.
Question I am after a solution for in particular:
Suppose I have the problem above where $f$ is smooth in all its variables and an initial value $I = (t_0, x(t_0), \dots, x^{(n-1)}(t_0))$.  Suppose further that for this initial value, I know that for any solution $x(t)$, $\|x(t)\| \leq g_I(t)$ where $g_I$ is a smooth function defined over all of $\Bbb R$.  Can I then conclude there is a unique solution to the IVP given by $I$ defined over all of $\Bbb R$ since $x(t)$ does not diverge in finite time (and there is local existence around $t_0$)?  Or do I require more, that all of the derivatives $$\| (x(t), x^{(1)}(t), \dots, x^{(n-1)}(t)) \| \leq h_I(t)$$
Thanks
 A: Following up on the comments - you are right:
Consider the initial value problem 
$$\dot{z}=g(t,z),\quad\quad z(t_0)=z_0\in\mathbb{R}^n$$
where $g$ is defined on $\mathbb{R}\times\mathbb{R}^n$ and $\mathcal{C}^1$. Note that you can re-write your problem as the above by setting
$$z:=(x,x^1,\dots,x^{n-1}),\quad\quad z_0=I, \quad\quad g_i(t,z):=z_{i+1}\quad\forall i\neq n,$$
$$g_n(t,z):=f(t,xz)$$
Then, Khalil states (in p. 93)
"There is a maximum interval $[t_0,T)$ where the unique solutions starting at $(t_0,z_0)$ exists. If $T<\infty$ then as $t\rightarrow T$, the solution leaves any compact set over which $f$ is locally Lipschitz in $x$".
Since $g$ is $\mathcal{C}^1$, it is Lipschitz at any point (that is, there is a neighbourhood around the point such that the Lipschitz condition holds), thus the Lipschitz condition holds over any compact set (just cover the set, extract a finite sub cover and pick the maximum Lipschitz constant). Then, by contradiction, you can conclude that $x$ exists over $[t_0,\infty)$.
Remark: He doesn't prove in the book what he says in the quote. He says a proof for the phrase bit can be found in section 8.5 of this or in section 2.3 of this - I suspect this is pretty standard and can be found in many other places as well. The rest he leaves as an exercise. If you'd like, let me know and I can (attempt) to prove.
