  We propose and study a one-dimensional $2\times 2$ hyperbolic Eulerian system
with local relaxation from critical threshold phenomena perspective. The system
features dynamic transition between strictly and weakly hyperbolic. For
different classes of relaxation we identify intrinsic critical thresholds for
initial data that distinguish global regularity and finite time blowup. For
relaxation independent of density, we estimate bounds on density in terms of
velocity where the system is strictly hyperbolic.
