Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law

This paper examines the large-time behavior of solutions to a one-dimensional conservation law featuring a non-convex flux and an artificial heat flux term regulated by Cattaneo's law, forming a 2$\times$2 system of hyperbolic equations. Under the conditions of small wave strength and sufficiently small initial perturbations, we demonstrate the time-asymptotic stability of a composite wave that combines a degenerate Oleinik shock and a rarefaction wave. The proof utilizes the Oleinik entropy condition, the a-contraction method with time-dependent shifts, and weighted energy estimates.