Proposition 12.3.3.label If $G,\p_{0},...,\p_{k-1}$ are $C^{\om}$, then the PDE has at most one $C^{\om}$ solution.
Proof. We assume $t=0$ to be non-characteristic, Cauchy problem becomes $\begin{cases}\pd_{t}^{k}u=G(x,t,\parens{\pd_x^\al\pd_t^ju}_{\abs{\al}+j\leq k,0\leq j\leq k-1})\\ \pd_{t}^{j}u(x,0)=\p_{j}(x),0\leq j\leq k-1\end{cases}$ Then $\pd_{t}^{k+1}u$ is determined on $t=0$ by the PDE and
\begin{align*}\pd_{t}^{k+1}u=\pder{G}{t}+\sums{}{\abs{\al}+j\leq k}\pder{G}{u_{\al j}}\pd_{x}^{\al} \pd_{t}^{j+1}u\end{align*}
By induction, we are done.$\square$
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