Global characteristic flow under linear growth

If $F\in C^1(\mathbb{R}\times {\mathbb{R}}^d)$ and $|F(t,x)|\le K(1+|x|)$, every characteristic curve is defined for all real time. The Gronwall inequality bounds $1+|X(t)|$ on each bounded time interval, so local ordinary differential equation existence cannot terminate through escape to infinity. Uniqueness and the variational equation give a global $C^1$ spatial flow diffeomorphism, with Jacobian $J>0$ satisfying $J_t=(\mathrm{div}F)(t,X)J$.