 Solution

For the first-order quasilinear equation $a(x,u)\cdot \nabla u=b(x,u)$, suppose the coefficients and initial data $u=g$ on a smooth hypersurface $\Sigma$ are $C^1$. At an initial point $x_0$, require $a(x_0,g(x_0))\cdot n(x_0)\ne 0$ for a normal $n$ to $\Sigma$; this is precisely the condition that $\Sigma$ be a non-characteristic hypersurface. Then in a neighborhood of $x_0$ there is a unique $C^1$ solution taking the given data. Smooth coefficients and data give corresponding higher regularity.

The method of characteristics solves $\dot{x}=a(x,u)$, $\dot{u}=b(x,u)$ from each initial point. The non-characteristic condition makes the map from the initial coordinates and characteristic time to $x$ locally invertible, so these characteristic solutions define the asserted function.