Consider the ODE
$$\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x))$$
$$\Phi(0,x) = x, \quad x \in \mathbb{R}^N.$$

Assume that $$\frac{|\boldsymbol{F}|}{1+|x|} \in L^1\left([0,T]; L^1(\mathbb{R}^N) \right) + L^1\left([0,T]; L^\infty(\mathbb{R}^N) \right).$$

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  
$$\Phi(t,x) \in C(t) \ ?$$


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