  In this paper, we define the rarefaction and compression characters for the
supersonic expanding wave of the compressible Euler equations with radial
symmetry. Under this new definition, we show that solutions with rarefaction
initial data will not form shock in finite time, i.e. exist global-in-time as
classical solutions. On the other hand, singularity forms in finite time when
the initial data include strong compression somewhere. Several useful invariant
domains will be also given.
