While studying the existence and uniqueness of the solutions of ODEs, I encountered a problem about local and global solutions, which is as the follows.
Consider $I=\mathbb{R}$ and F a $C^1$ vector field on $I\times \mathbb{R}^m$ such that $$\forall t\in\mathbb{R}, u\in\mathbb{R}^{m},\quad|\mathbf{F}(t,\mathbf{u})|\leq C(1+|\mathbf{u}|), \quad \text{for some constant } C>0.$$ Prove that the following ODE has solutions that are global: $$\frac{d\mathbf{u}}{dt}=\mathbf{F}(t,\mathbf{u}(t)), \quad t\in\mathbb{R}.$$
Here are some of my thoughts. Because $\mathbf{F}$ is $C^1,$ by the Cauchy-Peano theorem, we know that for any initial values $(t_0,\mathbf{u}_0)\in I\times\mathbb{R}^m$, the ODE has a local solution in the neighbourhood of $(t_0,\mathbf{u}_0)$. Therefore, it is enough to show that this solution is global.
But then I got stuck, and I'm not sure how to show the solution exist for all times. Is there any methods to solve this problem and prove that the solution is global?
Many thanks for all the hints and solutions!