= Solution
The total differential order of $\partial_t-\partial_x^2$ is two, so its <principal symbol> is $p(\tau,\xi)=-\xi^2$. The conormal to $t=0$ is $(1,0)$, on which $p$ vanishes. \b[The initial line is a <characteristic hypersurface>] for this total-order symbol; the first-order time derivative does not enter it.
Suppose a <real analytic> solution existed near $(0,0)$. Repeated use of the <heat equation> gives $\partial_t^k u=\partial_x^{2k}u$. The initial <power series> is $\sum_{j\geq0}(-1)^j x^{2j}$ near zero, hence
$$
\partial_t^k u(0,0)=(-1)^k(2k)!.
$$
The time <Taylor series> at $x=0$ would therefore have coefficients $(-1)^k(2k)!/k!$. The ratio of successive absolute coefficients is $2(2k+1)\to\infty$, giving radius of convergence zero. This contradicts the assumed <real analytic> regularity. \b[No such analytic local solution exists], although the initial function itself is <real analytic>.
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