This manuscript focuses on the study of the existence of a solution for an abstract Cauchy problem involving a wave equation with a monotone operator damping and nonlinear source term. We apply the potential well and prove the global weak solutions and the exponential stability for initial data in the set of stability created from the Nehari Manifold. A more general framework is presented focusing on the damping, which must satisfy the monotonicity condition, and then, some previous works to second-order differential equations of hyperbolic type can be considered as a particular case.