= Solution
Write
$$
P=\partial_t-\partial_x^2-\partial_y^2.
$$
As a differential operator of order two, its <principal symbol> is
$$
\sigma_2(P)(\tau,\xi,\eta)=-(\xi^2+\eta^2).
$$
If $\nu=(\nu_t,\nu_x,\nu_y)$ is a unit normal to the real-analytic hypersurface $\Sigma$, the <non-characteristic hypersurface> condition is therefore
$$
\boxed{\sigma_2(P)(\nu)\ne0
\quad\Longleftrightarrow\quad
\nu_x^2+\nu_y^2>0.}
$$
Under this condition the equation can be solved for the second derivative normal to $\Sigma$. The <Cauchy-Kovalevskaya theorem> then gives a unique real-analytic solution in a neighbourhood of $\Sigma$ for the prescribed analytic Cauchy data $u|_\Sigma$ and $\partial_\nu u|_\Sigma$.
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