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

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 radius of the ball $B_{R(t)}(0)$ such that $$\Phi(t,x) \in B_{R(t)}(0) \ ?$$