= Solution
For a first-order <quasilinear partial differential equation>, take $C^1$ coefficients $a_j,b$, $C^1$ prescribed data $u_0$ on a smooth $C^2$ <hypersurface> $S$, and a point $x_0\in S$. The data are non-characteristic there if
$$
a(x_0,u_0(x_0))\cdot n(x_0)\ne0,
$$
where $n$ is any nonzero normal to $S$. Then there is a neighborhood of $x_0$ on which a unique $C^1$ solution realizes the data. Uniqueness is local, after restricting to a neighborhood swept out by the nearby characteristics.
The <method of characteristics> solves $\dot x=a(x,u)$, $\dot u=b(x,u)$ from points of $S$. In local surface coordinates $\eta$, the map $(\eta,s)\mapsto x(\eta,s)$ has nonsingular derivative at $s=0$ exactly under the stated normal condition. The <inverse function theorem> makes these variables local coordinates, giving the asserted solution and uniqueness.
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