Erbay, Hüsnü AtaErbay, SaadetErkip, A.2020-09-042020-09-042019-051078-0947http://hdl.handle.net/10679/6893https://doi.org/10.3934/dcds.2019119We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. We prove a long-time existence result for the nonlocal wave equations with a power-type nonlinearity and a small parameter. As the energy estimates involve a loss of derivatives, we follow the Nash-Moser approach proposed by Alvarez-Samaniego and Lannes. As an application to the long-time existence theorem, we consider the limiting case in which the kernel function is the Dirac measure and the nonlocal equation reduces to the governing equation of one-dimensional classical elasticity theory. The present study also extends our earlier result concerning local well-posedness for smooth kernels to nonsmooth kernels.engrestrictedAccessarticle3952877289100045608770002210.3934/dcds.2019119Long-time existenceNonlocal wave equationNash-Moser iterationImproved Boussinesq equation2-s2.0-85061305651