 Solution

The characteristics solve ${\dot{x}}_j=a_j(x)$ and $\dot{u}=b(x,u)$, starting with $x$ on the initial hypersurface and $u=\phi (x)$. The non-characteristic hypersurface condition at a point is $a\cdot n\ne 0$, where $n$ is any nonzero normal there. It makes the map from initial-surface coordinates and characteristic time to $x$ locally invertible. Local uniqueness for the ordinary differential equations and the inverse function theorem then give a unique local $C^1$ solution near that point, for the stated $C^1$ coefficients and data.