Formation and construction of shock for p-system under degenerate conditions of finite or infinite orders

In this paper, we are concerned with the shock formation and construction for $p$-system when the initial data are degenerate with finite or infinite orders (see (1.6) or (1.7) below). Since the solution we consider is a simple wave before shock formation, then with the help of the construction of shock solutions for the corresponding scalar equation, we establish the detailed estimates of the approximate solutions and then prove the convergence of the approximate solutions. Thus a weak entropy solution and a shock curve starting from the blowup point are constructed under two types of degenerate conditions. Meanwhile, some precise descriptions on the behaviors of the solutions near the blowup point are given.