We study a traffic flow model with inhomogeneous road conditions
such as obstacles. The model is a system of nonlinear hyperbolic equations
with both relaxation and sources. The flux and the source terms depend on
the space variable. Waves for such a system propagate in a more complicated
way than those do for models with homogeneous road conditions.
The $L^1$ well-posedness theory for the model is established. In particular,
we derive the continuous dependence of the solution on its initial data in $L^1$
topology. Moreover, the $L^1$-convergence to the unique zero relaxation limit is
proved. Finally, the asymptotic states of a general solution whose initial data
tend to constant states as $|x| \rightarrow +\infty$ are constructed.