Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations

TL;DR

This paper proves long-time existence of solutions for a broad class of nonlocal nonlinear wave equations modeling bidirectional wave propagation in elastic media, using Nash-Moser techniques to handle derivative loss.

Contribution

It extends previous local well-posedness results to long-time existence for nonsmooth kernels and applies Nash-Moser methods to nonlocal wave equations with power nonlinearities.

Findings

Long-time solutions exist for nonlocal wave equations with small parameters.

The approach handles nonsmooth kernels extending earlier smooth kernel results.

Application to classical elasticity when the kernel is a Dirac measure.

Abstract

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.