OK, from the comments I understand that the problem boils down to
$$\frac{\partial}{\partial t}\Phi_2(t,x_1,x_2)=\theta\bigl(F[\Phi_2(t,x_1,x_2)]-x_1\bigr),\;\;\Phi_2(0,x_1,x_2)=x_2,$$
where $\theta$ is the unit step function. Let me assume that $F$ is a nondecreasing function. Then the solution is
$$\Phi_2(t,x_1,x_2)=\begin{cases} t+x_2&\text{if}\;\;F(x_2)>x_1\\ x_2&\text{if}\;\;F(x_2)\leq x_1. \end{cases} $$
A more general choice of $F$ can be readily accommodated, by piecing together increasing and decreasing segments.