Hi! Can somebody help me with this exercise?

Let $\displaystyle F: \mathbb{R} \rightarrow \mathbb{R}^n, \, g: \mathbb{R}^n \rightarrow \mathbb{R}$ be smooth. Consider the following IVP on $\displaystyle \mathbb{R}^n \times (0, \infty):$

$\displaystyle u_t+F'(u) \cdot Du=0$

$\displaystyle u(x,0)=g(x)$

Under which condition on $\displaystyle F$ and $\displaystyle g$ does $\displaystyle u(x,t)=g(x-tF'(u(x,t)))$ define an implicit solution? When does this condition fail?

I am pretty aimless and would be thankful for any ideas.