Remark 12.3.9.label It may happen that the PDE have more than one solution in the sense that there exists non-analytic (say $C^{\infty}$) solution in addition to the $C^{\om}$ solution.

If there are points where $\pder{F}{\pd{\pd_t^k u}}=0$ there is a version of Cauchy-Kovalevskaya called the Ramified C-K theorem proved by J-Leray (1970’s) Garding wrote something on this.