Semilinear and quasilinear first-order Cauchy problems on a parametrised hypersurface

Let $n\ge 1$ and $V\subseteq {\mathbb{R}}^{n-1}$ be open, let $\gamma :V\to {\mathbb{R}}^n$ and $\varphi :V\to \mathbb{R}$ be $C^1$, with $\mathrm{rank}D\gamma (y)=n-1$ for every $y\in V$, and let $a:O\to {\mathbb{R}}^n$ and $b:O\to \mathbb{R}$ be smooth on an open set $O\subseteq {\mathbb{R}}^n\times \mathbb{R}$. Assume that $(\gamma (y),\varphi (y))\in O$ for every $y\in V$. The Cauchy problem for the quasilinear equation is

$a(x,u(x))\cdot Du(x)=b(x,u(x)),u(\gamma (y))=\varphi (y)\mathrm{.}$

It is semilinear when $a=a(x)$ is independent of $u$. At $y_0\in V$, a classical local solution is a $C^1$ function $u$ on an open neighbourhood $\mathrm{\Omega }$ of $\gamma (y_0)$ such that $(x,u(x))\in O$ and the PDE holds for every $x\in \mathrm{\Omega }$, and such that there is a neighbourhood $W\subseteq V$ of $y_0$ with $\gamma (W)\subseteq \mathrm{\Omega }$ and $u(\gamma (y))=\varphi (y)$ for every $y\in W$. This specializes the first-order classification in Linear, semilinear, quasilinear, and fully nonlinear partial differential equations; the word noncharacteristic will be tested by the rank calculation below, rather than by importing the space-time transport convention of Noncharacteristic Cauchy surfaces for first-order transport.