  We prove existence and boundedness of classical solutions for a family of
viscous conservation laws in one space dimension for arbitrarily large time.
The result relies on H. Amann's criterion for global existence of solutions and
on suitable uniform-in-time estimates for the solution. We also apply
J\"ungel's boundedness-by-entropy principle in order to obtain global existence
for systems with possibly degenerate diffusion terms. This work is motivated by
the study of a physical model for the space-time evolution of the strain and
velocity of an anharmonic spring of finite length.
