Hyperbolic Damped $p$-System and Diffusion Phenomena

In this survcy paper, wc review the dcvelopmcnt and progress of the study on the $2\cross 2$ hyperbolic p-system with damping. The damping cffort makes such a systcm to behave as a diffusion equation. The focus in this paper is to show how to find the best asymptotic profile for the dampcd p-system, and what arc the optimal convcrgent ratcs. Thc most new results arc reported in this paper.