This paper is concerned with the nonlinear stability of traveling wave
solutions for a quasi-linear relaxation model with a nonconvex
equilibrium flux. The study is motivated by and the results are applied
to the well-known dynamic continuum traffic flow model, the Payne and
Whitham (PW) model with a nonconcave fundamental diagram. The PW model
is the first of its kind and it has been widely adopted by traffic
engineers in the study of stability and instability phenomena of traffic
flow. The traveling wave solutions are shown to be asymptotically stable
under small disturbances and under the sub-characteristic condition using a
weighted energy method. The analysis applies to both non-degenerate case
and the degenerate case where the traveling wave has exponential decay
rates at infinity and has an algebraic decay rate at infinity, respectively.