Abstract: In this paper, we extend the result in[16] to general p(v). We prove that, under condition (M) when P ≥ 3/2, where P=ṗp/p, there exists a unique global continuous solution to the Riemann problem (E), (R), whose structure is similar to the local solution. When 1<P_* ≤ P^*<5/4, or P_*=P^*=5/4, or 5/4<P^* ≤ P^*<3/2, where P_*=inf P and P^*=sup P for all u under consideration, if at least one of the initial centered rarefaction waves is sufficiently strong, then the solution must be breakdown in a finite time.