This paper is devoted to the study of a nonlinear Kirchhoff-Carrier wave equation in an annular associated with nonhomogeneous Dirichlet conditions. At first, by applying the Faedo-Galerkin, we prove existence and uniqueness of the solution of the problem considered. Next, by constructing Lyapunov functional, we prove a blow-up result for solutions with a negative initial energy and establish a sufficient condition to obtain the exponential decay of weak solutions.