Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 3 / b / ii / Solution

One prescribes analytic Cauchy data: the value of $u$ and one first derivative transverse to $\Sigma$, for examplewhere $\varphi$ and $\psi$ are real analytic on $\Sigma$. Being a non-characteristic hypersurface allows the equation to solve for the second derivative in the transverse direction. The Cauchy-Kovalevskaya theorem then gives one and only one local real analytic solution near $p$.