  We prove that, for first-order, fully nonlinear systems of partial
differential equations, under an hypothesis of ellipticity for the principal
symbol, the Cauchy problem has no solution within a range of Sobolev indices
depending on the regularity of the initial datum. This gives a new and greatly
detailed proof of a result of G. M\'etivier [{\it Remarks on the Cauchy
problem}, 2005]. We then extend this result to systems experiencing a
transition from hyperbolicity and ellipticity, in the spirit of recent work by
N. Lerner, Y. Morimoto, and C.-J. Xu, [{\it Instability of the
Cauchy-Kovalevskaya solution for a class of non-linear systems}, 2010], and N.
Lerner, T. Nguyen and B. Texier [{\it The onset of instability in first-order
systems}, 2018].

Analysis of PDEs