Let $\dot{x}(t) \in F(x(t))$ be a differential inclusion, with $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ an uppersemicontinuous, convex and compact valued set-valued map.

On Wikipedia it is said that such an inclusion always admits a local solution (i.e. on an interval $[0, \epsilon)$ ), and if such a solution does not blow up, then we can extend it to a global solution (i.e. on $[0, +\infty)$ ). Such a result seems quite intuitive, and I believe has a simple proof.

Is there any reference where such a result is explicitly stated? I have only managed to found some reference which impose conditions on $F$.

Any help would be appreciated.

EDIT: Actually, looking at the proof of (local) existence in Aubin-Cellina, I am no longer sure that this is true. However, I am not able to construct a counterexample.