In this paper, the wave equation defined in a semi-infinite cylinder is considered,
in which the nonlinear damping and source terms is included. By setting an arbitrary
parameter greater than zero in the energy expression, the fast growth rate or decay rate
of the solution with spatial variables is obtained by using energy analysis method and
differential inequality technique. Secondly, we obtain the asymptotic behavior of the
solution on the external domain of the sphere. In addition, in this paper we also give
some useful remarks which show that our results can be extended to more models.