Sharp critical thresholds in a hyperbolic system with relaxation

<p style='text-indent:20px;'>We propose and study a one-dimensional <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The features dynamic transition between strictly weakly hyperbolic. For different classes of we identify intrinsic <b>critical thresholds</b> for initial data that distinguish global regularity finite time blowup. independent density, estimate bounds on density in terms velocity where the is hyperbolic.</p>