We study the large-time behavior of the solution to an initial boundary value
problem on the half line for scalar conservation law, where the data on the
boundary and also at the far field are prescribed. In the case where the flux is
convex and the corresponding Riemann problem for the hyperbolic part admits the
transonic rarefaction wave (which means its characteristic speed changes the
sign), it is known by the work of Liu-Matsumura-Nishihara (’98) that the
solution tends toward a linear superposition of the stationary solution and the
rarefaction wave of the hyperbolic part. In this paper, it is proved that even
for a quite wide class of flux functions which are not necessarily convex, such
the superposition of the stationary solution and the rarefaction wave is
asymptotically stable, provided the rarefaction wave is weak. The proof is given
by a technical $L^2$-weighted energy method.