Growth assumption and example of finite (arbitrarily small) time blow up for ODE

Consider the following ODE initial value problem \begin{align*} &\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\ &\Phi(0,x) = x, & x \in \mathbb{R}^N. \end{align*}

We say that $ \Phi: [0,T] \times \mathbb{R}^N \to \mathbb{R}^N$ is the flow of the ODE (as in this paper) if it solves it in some sense.

We assume that the vector field $ \boldsymbol{F}:[0,T]\times \mathbb{R}^N \to \mathbb{R}^N$ is Sobolev and such that that $ $ (*) \qquad \frac{|\boldsymbol{F}|}{1+|x|} \in L^1\left([0,T]; L^1(\mathbb{R}^N) \right) + L^1\left([0,T]; L^\infty(\mathbb{R}^N) \right),$ $ that is, there exist \begin{align*} &\boldsymbol{F}_1 \in L^1\left([0,T]; L^1(\mathbb{R}^N) \right)\ &\boldsymbol{F}_2 \in L^1\left([0,T]; L^\infty(\mathbb{R}^N) \right) \end{align*} such that $ $ \frac{\boldsymbol{F}}{1+|x|} = \boldsymbol{F}_1 + \boldsymbol{F}_2.$ $

In an answer to Quantitative finite speed of propagation property for ODE (cone of dependence), it has been remarked that the flow $ \Phi$ can blow up in finite (and arbitrarily small) time if the $ F_1\neq 0$ .

  1. Can you provide an example of such flow that blows up in finite (and arbitrarily small) time?

  2. Why is this not in contrast with the fact that assumption (*) is used in the existence and uniqueness result of Theorem 30 (page 23) of this paper?

  3. In the theorem cited in the previous point, is assumption (*) key for existence or uniqueness?