On hyperbolicity and Gevrey well-posedness. Part two: Scalar or degenerate transitions

For first-order quasi-linear systems of partial differential equations, we formulate an assumption of a transition from initial hyperbolicity to ellipticity. This assumption bears on the principal symbol of the first-order operator. Under such an assumption, we prove a strong Hadamard instability for the associated Cauchy problem, namely an instantaneous defect of H\"older continuity of the flow from $G^{\sigma}$ to $L^2$, with $0 < \sigma < \sigma_0$, the limiting Gevrey index $\sigma_0$ depending on the nature of the transition. We restrict here to scalar transitions, and non-scalar transitions in which the boundary of the hyperbolic zone satisfies a flatness condition. As in our previous work for initially elliptic Cauchy problems [B. Morisse, \textit{On hyperbolicity and Gevrey well-posedness. Part one: the elliptic case}, arXiv:1611.07225], the instability follows from a long-time Cauchy-Kovalevskaya construction for highly oscillating solutions. This extends recent work of N. Lerner, T. Nguyen, and B. Texier [\textit{The onset of instability in first-order systems}, to appear in J. Eur. Math. Soc.].