  Motivated by radiation hydrodynamics, we analyse a 2x2 system consisting of a
one-dimensional viscous conservation law with strictly convex flux -- the
viscous Burgers' equation being a paradigmatic example -- coupled with an
elliptic equation, named viscous Hamer-type system. In the regime of small
viscosity and for large shocks, namely when the profile of the corresponding
underlying inviscid model undergoes a discontinuity -- usually called sub-shock
-- it is proved the existence of a smooth propagating front, regularising the
jump of the corresponding inviscid equation. The proof is based on Geometric
Singular Perturbation Theory (GSPT) as introduced in the pioneering work of
Fenichel [5] and subsequently developed by Szmolyan [19]. In addition, the case
of small shocks and large viscosity is also addressed via a standard
bifurcation argument.
