<p style="text-indent:20px;">This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the <inline-formula><tex-math id="M1">\begin{document}$ L^2\cap L^\infty- $\end{document}</tex-math></inline-formula>norm of the initial perturbation is assumed to be small, the <inline-formula><tex-math id="M2">\begin{document}$ H^1- $\end{document}</tex-math></inline-formula>norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy method.</p>

In this paper, we mainly consider Riemann problem for the widely used nonsimplified chromatography system with initial data consisting of three pieces of constant states. Through phase plane analysis, the solutions of the nonsimplified chromatography system are established. When the different initial data tend to −1 from the right side, the existence of zero shock wave, zero delta shock wave, and zero rarefaction wave is obtained via analyzing its wave interaction. Finally, the correctness of the main conclusions is verified by numerical simulation, and the numerical results are in good agreement with the theoretical solutions of several experimental cases.