Let $F:\mathbb{R}^2 \to \mathbb R$ be a bounded Lipschitz function and $G(x,y) = \chi_{\{x \le F(y)\}}$. 

Consider the ODE 
$$
\begin{cases}
\partial_t \Phi(t,x) = G(\Phi), & t \in [0,T]\\
\Phi(0,x) = x & x \in \mathbb R^2
\end{cases}
$$

 - How can we write the solution $\Phi$ explicitly?