Let $n>1$ and $u$ be a solution of a linear PDE with constant coefficients $$ u_t-\sum_{k=0}^n a_k \partial_x^k u=0,\quad a_k\in \mathbb C,\quad a_n\ne0, $$ in some neighborhood of a point $(x_0,t_0)$ on a plane. Is it true that the function $u(x,t_0)$ is analytic in some neighborhood of $x_0$?

In some special cases, such as the heat equation it follows from results of Petrovskii, namely that solutions of homogeneous parabolic equations with analytic coefficients are analytic with respect to space variables. But in the general case I'm unable to find a reference.