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This study deals with the analysis of the Cauchy problem of a general classof nonlocal nonlinear equations modeling the bi-directional propagation ofdispersive waves in various contexts. The nonlocal nature of the problem isreflected by two different elliptic pseudodifferential operators acting onlinear and nonlinear functions of the dependent variable, respectively. Thewell-known doubly dispersive nonlinear wave equation that incorporates twotypes of dispersive effects originated from two different dispersion operatorsfalls into the category studied here. The class of nonlocal nonlinear waveequations also covers a variety of well-known wave equations such as variousforms of the Boussinesq equation. Local existence of solutions of the Cauchyproblem with initial data in suitable Sobolev spaces is proven and theconditions for global existence and finite-time blow-up of solutions areestablished.

Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations