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

H A Erbay,A Erkip,S Erbay

doi:10.3934/dcds.2019119

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.

Highlights

In the present paper we prove the long-time existence and uniform estimates of solutions to the Cauchy problem utt = β ∗ u + ǫpup+1 xx, x ∈ R, t > 0, (1)
The nonlocal wave equation (1) describes the one-dimensional motion of a nonlocally and nonlinearly elastic medium and u represents the elastic strain
If β is taken as the Dirac measure, (1) reduces to the nondispersive nonlinear wave equation utt − uxx = ǫp up+1 xx of classical elasticity

Summary

Introduction

In the present paper we prove the long-time existence and uniform estimates of solutions to the Cauchy problem utt = β ∗ u + ǫpup+1 xx, x ∈ R, t > 0,. To prove our long-time existence result we start by converting (1) into a perturbation of the symmetric hyperbolic linear system and obtain the energy estimates for the corresponding linearized equation. In addition to the studies about asymptotic models of water waves, there are studies presenting the rigorous derivation of various asymptotic models for nonlinear elastic waves in the long-wave-small-amplitude regime (for instance we refer the reader to [5] where the Camassa-Holm equation and (1) are compared). The long-time existence result and uniform bound obtained in this study will be used in a future work to explore comparison of nonlocal equations.

Preliminaries

Energy estimates for a related linear system

Hs from which it follows that

Preparation for the Nash-Moser scheme

Long-time existence of solutions