Definition 12.3.1 (Setting and Non-Characteristics).label Suppose
\begin{align*}\begin{cases}F(x,\parens{\pd^\al u}_{\abs{\al}\leq k})=0\\ \pd_{\nu}^{j}u|_{S}=\p_{j}&0\leq j\leq k-1\end{cases}\end{align*}
such that
\begin{align*}\begin{cases}F\in C^{\om}\\ s\in C^{\om}\end{cases}\end{align*}
then when $\exists! C^{\om}$ solution $u$ in a neighborhood $\OM$? By constant rank theorem, we can change coordinates so that locally, $S$ is given by $t=0$, $(\UBC{x_1,...,x_{n-1}}{x},t)$ coordinates. The Cauchy problem then becomes
\begin{align*}\begin{cases}F(x,t,\parens{\pd_x^\al\pd_t^ju}_{\abs{\al}+j\leq k})=0\\ \parens{\pd_t^ju}(x,0)=\p_{j}(x)&1\leq j\leq k-1\end{cases}\end{align*}
We remark that Cauchy data determine all derivatives $\parens{\pd_x^\al\pd_t^j}u(x,0)$ as long as $j\leq k-1$. So the only derivative of $u$ which are unknown on $S$ are $\parens{\pd_t^ku}(x,0)$.
$S$ (given by $t=0$) is non-characteristic for
\begin{align*}\begin{cases}F(x,t,\parens{\pd_x^\al\pd_t^ju}_{\abs{\al}+j\leq k})=0\\ \parens{\pd_t^ju}(x,0)=\p_{j}(x)&1\leq j\leq k-1\end{cases}\end{align*}
if one can solve for $\pd_{t}^{k}u$.
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