My lectures notes on ODE mention the following theorem:
Proposition (Lifespan): Let $I \subseteq \mathbb{R}$ be an interval, $D \subseteq I \times \mathbb{R}$ open, $(t_0,y_0) \in D$ and $f: D \to \mathbb{R}$ continuous and locally Lipschitz with respect to $y$. Let $U(t_0,y_0)$ be the neighbourhood of $(t_0,y_0)$ from the local Lipschitz condition and $h,c>0$ and define
$$Q:=[t_0,t_0+h] \times [y_0-c,y_0+c] \subseteq U(t_0,y_0),$$
then the maximal interval of existence of the solution to the IVP
$\left\{\begin{array}{} y'(t)=f(t,y(t)), t \in I\\ y(t_{0})=y_{0}\\ \end{array} \right.$
contains $[t_0,t_0+a]$ where $a=min\{h,c/M\}$ with $M=\max \limits_{(t,y) \in Q} \{\lvert f(t,y) \rvert\}$.
Proof: We define the function
$\tilde{f}(t,y) = \left\{\begin{array}{} f(t,y_0-c), t \in [t_0,t_0+h], y<y_0-c\\ f(t,y), t \in [t_0,t_0+h], y_0-c<y<y_0+c\\ f(t,y_0-c), t \in [t_0,t_0+h], y_0+c<y\\ \end{array} \right.$
then for all $(t,y) \in [t_0,t_0+h] \times \mathbb{R}$ the argument of $\tilde{f}$ lies in $Q$, and thus in $U(t_0,y_0)$. So $\tilde{f}$ satisfies a global Lipschitz condition on $[t_0,t_0+h]$, so the IVP
$\left\{\begin{array}{} y'(t)=\tilde{f}(t,y(t)), t \in [t_0,t_0+h]\\ y(t_{0})=y_{0}\\ \end{array} \right.$
has a unique solution $v$ on $[t_0,t_0+h]$. So for $t \in [t_0,t_0+a]$ (note that $a \leq h$)
$$\lvert v(t)-y_0 \rvert=\lvert y_0 + \int \limits_{t_0}^{t} v'(s) ds > - y_0\vert=\lvert \tilde{f}(s,v(s)) ds \rvert.$$
In addition, $\tilde{f}$ is bounded by $M$ by construction, so we have
$$\lvert v(t) - y_0 \rvert \leq M \lvert t-t_0\rvert \leq Ma \leq M > \frac{c}{M}=c$$
It follows that $v(t) \in [y_0-c,y_0+c]$ for all $t \in [t_0,t_0+a]$. But this means that for all $t \in [t_0,t_0+a]$, we have $\tilde{f}(t,v(t))=f(t,v(t))$, and so
$$v'(t)=\tilde{f}(t,v(t))=f(t,v(t))$$
and $v(t_0)=t_0$. But then $v$ is a solution to the IVP given in the theorem on $[t_0,t_0+a]$. So the maximal interval of existence must contain at least $[t_0,t_0+a]$.
The author also states that this can be generalized to systems of ODEs, but I could not find anything on google. The idea of the proof for a single ODE is to extend the function to a globally Lipschitz function by changing its values outside of the cube $Q$. My guess is that the same line of argument works for systems, i.e. we need to choose suitable values $h,c_1,...,c_d$ such that the solution does not leave the cube $Q$ in any direction. I think we can restrict ourselves to directions parallel to the axes since these are the shortest paths to get outside of $Q$. However, I cannot see how to define a globally Lipschitz function $g$ that agrees with $f$ on $Q$ in the case of a $d$-dimensional system.
Could anyone please point me to a source that proves this theorem for systems or give a proof for this case?
Thanks a lot!
Define $$ \psi(y)=y_0+\frac{c}{\max(c,\|y-y_0\|)}(y-y_0). $$ This is a continuous function with $\psi(y)=y$ for $\|y-y_0\|\le c$ and constant on rays through $y_0$ outside this "ball".
Now define $\tilde f(t,y)=f(t,\psi(y))$ and continue in the same fashion as before.
One might have to use $$ \left\|\frac{a}{\|a\|}-\frac{b}{\|b\|}\right\| \le\left\|\frac{a}{\|a\|}-\frac{a}{\|b\|}\right\|+\left\|\frac{a}{\|b\|}-\frac{b}{\|b\|}\right\| =\|a\|\,\frac{|\|b\|-\|a\||}{\|b\|\,\|a\|}+\frac{\|a-b\|}{\|b\|} \le\frac{2}{\|b\|}\|a-b\| $$ or better to prove the global Lipschitz property.