Existence of solution for the ODE $x' = F(x)$ when $F$ is a $C^\infty$ function

Consider the system of differential equations of form $$x'(t) = F(x(t))$$ where $x(t) \in \mathbb{R}^n$ and the function $F:\mathbb{R}^n \to \mathbb{R}^n$ is a $C^\infty$ function.

I am looking for results on existence of solutions on compact intervals defined on time interval $ t \in [0,T]$ for all $T > 0$. Any references are appreciated.