  We 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.
