Existence and Non-Existence of Global Smooth Solutions for p-System with Relaxation

In this paper, we consider the Cauchy problem for p-system with relaxation. Under the assumption that the relaxation time is sufficiently small, we prove the existence of the global smooth solution to the Cauchy problem with C1-initial data provided the C0-norm of the derivative of the initial data is of the order of ξ/ . Here ξ is a small positive constant. On the other hand, when the initial density has compact support but is not identically zero, we prove the global regular solution for the Cauchy problem does not exist.