  This study deals with the analysis of the Cauchy problem of a general class
of nonlocal nonlinear equations modeling the bi-directional propagation of
dispersive waves in various contexts. The nonlocal nature of the problem is
reflected by two different elliptic pseudodifferential operators acting on
linear and nonlinear functions of the dependent variable, respectively. The
well-known doubly dispersive nonlinear wave equation that incorporates two
types of dispersive effects originated from two different dispersion operators
falls into the category studied here. The class of nonlocal nonlinear wave
equations also covers a variety of well-known wave equations such as various
forms of the Boussinesq equation. Local existence of solutions of the Cauchy
problem with initial data in suitable Sobolev spaces is proven and the
conditions for global existence and finite-time blow-up of solutions are
established.
