Thresholds for global existence and blow-up in a general class of doubly dispersive nonlocal wave equations

In this paper we study the global existence and blow-up of solutions for a general class of nonlocal nonlinear wave equations with power-type onlinearities, utt − Luxx = B(−|u|^(p−1)u)_xx, (p > 1), where the nonlocality enters through two pseudo-differential operators L and B. We establish thresholds for global existence versus blow-up using the potential well method which relies essentially on the ideas suggested by Payne and Sattinger. Our results improve the global existence and blow-up results given in the literature for the present class of nonlocal nonlinear wave equations and cover those given for many well-known nonlinear dispersive wave equations such as the so-called double-dispersion equation and the traditional Boussinesq-type equations, as special cases