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.
"Large-time Behavior of Solutions to an Initial-boundary Value Problem on the Half Line for Scalar Viscous Conservation Law." Methods Appl. Anal. 14 (1) 45 - 60, March 2007.