We study the properties of the solutions of scalar conservation laws with a source term, assuming that the flux is convex and that the initial value has compact support. We show that their asymptotic profile consists in a number of rarefaction waves divided by regions where the solutions oscillate around an unstable zero of the source term.
Carlo Sinestrari. "Asymptotic profile of solutions of conservation laws with source." Differential Integral Equations 9 (3) 499 - 525, 1996. https://doi.org/10.57262/die/1367969968