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

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.

Comparison of nonlocal nonlinear wave equations in the long-wave limit

“…This shows that for varying kernels with the same dispersive nature, solutions of (1) approximate each other and the approximation errors originate from the dispersive nature of the kernels rather than from their shapes. We note that, in the terminology of some authors, our long-time existence results [2] together with the work presented here are in fact consistency, existence, convergence results for the nonlocal bidirectional approximations of the nonlocal equation (1). We refer to [3][4][5][6][7] and the references therein for a detailed discussion of these concepts.…”

“…Clearly, when u 1 = (w 0 ) x with v 0 = K −1 δ w 0 the initial-value problem (1)-(2) becomes equivalent to (11)-(12); hence we have long time existence and uniform bounds for the solutions u ǫ,δ , v ǫ,δ of (11)- (12). In fact the long-time existence result in [2] was first proved for (11)- (12) and then extended to (1)-(2).…”

“…The computation in [2] shows that the optimal values of D and P are approximately 7.35 and 55.34, respectively. Throughout this work we will assume that the kernel β(x) is an even function with β(x)dx = 1 and satisfies (7).…”

“…(C1) β (1) and β (2) satisfy the ellipticity and boundness condition (7), (C2) β (1) and β (2) have the same first (2k − 1)-order moments, namely…”

On the convergence of the nonlocal nonlinear model to the classical elasticity equation

“…A collection of articles [5,[7][8][9][10][11] focuses on various aspects of mathematical analysis of the initial-value problems associated with (1.1). In those studies it is assumed that the Fourier transform β of the kernel function β satisfies the ellipticity and boundedness 2 condition…”