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For first-order quasi-linear systems of partial differential equations, weformulate an assumption of a transition from initial hyperbolicity toellipticity. This assumption bears on the principal symbol of the first-orderoperator. Under such an assumption, we prove a strong Hadamard instability forthe associated Cauchy problem, namely an instantaneous defect of H\"oldercontinuity 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 thetransition. We restrict here to scalar transitions, and non-scalar transitionsin which the boundary of the hyperbolic zone satisfies a flatness condition. Asin our previous work for initially elliptic Cauchy problems [B. Morisse,\textit{On hyperbolicity and Gevrey well-posedness. Part one: the ellipticcase}, arXiv:1611.07225], the instability follows from a long-timeCauchy-Kovalevskaya construction for highly oscillating solutions. This extendsrecent work of N. Lerner, T. Nguyen, and B. Texier [\textit{The onset ofinstability in first-order systems}, to appear in J. Eur. Math. Soc.].

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