Denote the real line by $\mathcal{R}$ and the non-negative real line by $\mathcal{R}^+$. Consider given functions $f:\mathcal{R}^n\times\mathcal{R}^m\longrightarrow\mathcal{R}^n$ and $u:\mathcal{R}^+\longrightarrow\mathcal{R}^m$. We then have the ODE system with input $u$: $$x'(t)=f(x(t),u(t)),$$ for all $t\in\mathcal{R}^+$. The solutions to this system are functions $x:\mathcal{R}^+\longrightarrow\mathcal{R}^n$ which are differentiable an satisfy the systems equation globally.
My question: Under which conditions on the function $f$ and the input $u$ can I claim that there exists at least a solution $x$ on all $\mathcal{R}^+$?