In this paper, we consider the Dirichlet problem for a wave equation of Kirchhoff- Carrier type with a nonlinear viscoelastic term. It consists of two main parts. In Part 1, we establish existence and uniqueness of a weak solution by applying the Faedo- Galerkin method and the standard arguments of density corresponding to the regularity of initial conditions. In Part 2, we give a sufficient condition for the global existence and exponential decay of the weak solutions by defining a modified energy functional together with the technique of Lyapunov functional.