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. 

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

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If $x \in B_{R}(0)$, what is the  [truncated cone with base $B_R(0)$][1], which we shall call $C(T)$, such that  
$$\Phi(t,x) \in C(T) $$
for all $t \in [0,T]$.


  [1]: http://web.tecnico.ulisboa.pt/~mcasquilho/compute/misc/,cone_vol/truncConeVert.jpeg