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.