I am looking for a general theorem that guarantees the existence of a global solution for an ODE system in $\mathbb{R}^n$


\begin{equation}
\left\{ \begin{aligned}
x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\
x(a) &= x\_0
\end{aligned} \right.
\end{equation}


By "global" I mean that the time interval is fixed, i.e. $[a,b]$, but I am not asking the solution to stay in an a priori fixed compact set of $\mathbb{R}^n$ (though the final solution will be absolutely continuous and thus bounded).
The setting is that of a possibly discontinuous vector field, described by the Carathéodory conditions, that is


1. $x \mapsto f(t,x)$ is continuous for a.e. $t$
2. $t \mapsto f(t,x)$ is measurable for each $x$
3. $|f(t,x)| \leq m(t)$, $m(t)$ being summable


A classical Carathéodory existence theorem (see e.g. Filippov, "Differential Equations with Discontinuous Right-Hand Side" (1988)) gives a local existence result in a compact set $K \subset \mathbb{R}^n$ under the above Charathéodory conditions.


Another classical Carathéodory theorem gives instead the global existence and uniqueness under a further Lipschitz continuity assumption:


4. $|f(t,x)-f(t,y)| \leq L(t) |x-y|$, $L(t)$ being summable


Finally, I found a global existence theorem (see Theorem II.3.2 on Reid, "Ordinary Differential Equations" (1971)), under the assumption


5. $|f(t,x)| \leq M(t)(1+|x|)$, $M(t)$ being summable


This last result require the vector field to have an at most linear growth in the variable $x$.
I was wondering if anyone knows more general results for the existence of a global solutions, which can include also more than linear growth, or if the results I quoted are already the best I can get.


Thank you!


C/p from [Math.StackExchange](https://math.stackexchange.com/questions/4060180/what-is-the-most-general-carath%C3%A9odory-type-global-existence-theorem).



What is the most general Carathéodory-type global existence theorem?
(N.B. In the below I assume $[a,b] = [0,\infty]$, but the precise values don't matter and appropriate substitutions of $a,b$ into the discussion also gives you the same conclusion.)


Once you have a local existence theorem of the form



> 
> For every compact set $K$ and compact subset $K\_0 \Subset K$, there exists $T > 0$ such that for every $x\_0\in K\_0$ there exists a solution to the IVP with initial value $x\_0$ defined on $[0,T]$ such that the trajectory remains in $K$ for all $T$
> 
> 
> 


Then upgrading to global existence is simply an issue of proving that any solution cannot escape to infinity in finite time. This is evidenced by the 5th theorem you cited: if you have
$$ |x'| \leq M(t) (1+|x|) $$
then
you have
$$ \frac{|x|'}{1+|x|} \leq M(t) $$
and integrating both sides you find
$$ \ln (1+|x|) \Big|\_{t = 0}^{t= T} \leq C $$
using that $M(t)$ is summable.


Slight refinement.
------------------


Let's abstract the argument a bit. Assume you have a bound on $f$, such that you can write


$$ |x'| \leq M(t) g(|x|) $$


for some given function $g:\mathbb{R}\to \mathbb{R}$ that is locally integrable. Denote by $G$ any primitive of the function $1/g$. Then integrating this differential inequality for a solution guarantees


$$ G(|x|) \Big|\_{t = 0}^{t = T} \leq C$$


provided that $M(t)$ is summable. And so **as long as $G$ is coercive** (in other words, it is [proper](https://en.wikipedia.org/wiki/Proper_map)) then the same argument as above will guarantee that $|x|$ does not blow-up in finite time and hence you have global existence.


You can therefore do slightly better then $(1+|x|)$, since there are superlinear $g$s for which $1/g$ is not integrable on the entire real line. For example, you may wish to take


$$ g(|x|) = (1 + |x|) \ln (1+|x|) $$


Then you have a primitive


$$ G(s) = \ln \ln (1+s) $$


which is still proper. In fact you can extend this using the usual tower of natural logs.


Negative examples
-----------------


But you can certainly not go beyond logarithmic improvements, at least in general.
You can see this easily by considering scalar equations of the form


$$ x' = M(t) (1 + |x|)^{1+\epsilon} $$


where $M$ is a positive summable function.


Note that with this assumption the solution is increasing, and so positive initial data will lead to positive solutions. And so if $x(0) = x\_0$ is positive, the solution satisfies


$$ \frac{x'}{(1+x)^{1+\epsilon}} = M(t) $$


which you can integrate to find


$$ \frac{1}{(1+x(T))^\epsilon} = \frac{1}{(1+x\_0)^\epsilon} - \epsilon \int\_0^T M(t) ~dt $$


And so with $M(t)$ considered fixed, for every $T$ there exists some $R$ such that if $x\_0 > R$ then $x$ must blow up prior to $t = T$.


Small data regime
-----------------


The previous discussion however hints at a small-data version of the result. Those that for the previous scalar equation, if


$$ (1 + x\_0)^\epsilon \leq \frac{1}{\epsilon \int\_0^\infty M(t) ~dt} $$


then we guarantee that $(1 + x(t))^{-1}$ is bounded away from infinity and cannot blow-up in finite time. And hence we have a statement of the form:



> 
> There exists a compact set $K\_0$ with non-empty interior such that for all $x\_0 \in K\_0$ there exists a (future-in-time) global solution.
> 
> 
> 


In the general form we gave earlier: let's assume without loss of generality that $g(|x|)$ is positive, and hence $G(s)$ is either unbounded (in which case you have that it is proper and global existence for all data) or that $\lim\_{s\to \infty} G(s) = G\_\infty$ exists.


In the latter case, let $C = \int\_0^\infty |M(t)|$. Then provided the set
$$ \{ r : G(r) < G\_\infty - C \} $$
is non-empty, any initial data $x\_0$ with $|x\_0|$ in this set will generate a solution that exists globally.


Additional Structures
---------------------


Note also that you may have a situation where you have additional structures. For example, it is possible that the best possible absolute value bound
$$ |f(t,x)| \leq M(t) (1+|x|)^\kappa $$
has $\kappa > 1$, but still you have global existence for all initial data. This would be the case if you happen to have
$$ x\cdot f(t,x) \leq 0 $$
in which case $|x|^2$ is (weakly) a Lyapunov function.


