# What is the most general Carathéodory-type global existence theorem?

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

\left\{ \begin{aligned} x'(t) &= f(t, x(t)), \qquad t \in [a,b] \\ x(a) &= x_0 \end{aligned} \right.

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:

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

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

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

• Just look at the scalar case $\dot{x} = x^{1+\epsilon}$ with positive initial data: If you allow $f$ to grow superlinearly in $x$, you have finite time blow-up solutions. By making your initial data large you can make the blow-up arbitrarily fast. And so summability of some $M(t)$ will not help you there. Mar 27, 2021 at 0:39
• OTOH, if you are willing to put in smalleness assumptions on the initial data, then there can be a competition. Mar 27, 2021 at 0:40
• Thank you. What about the smallness assumptions on the initial data? Could you provide some examples? Mar 27, 2021 at 11:46

(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) 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.

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.