The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials

TL;DR

This paper analyzes the Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations modeling anti-plane shear in elastic materials, establishing conditions for global existence or finite-time blow-up of solutions.

Contribution

It provides new analytical results on the existence and blow-up behavior of solutions for nonlocal nonlinear wave equations in elasticity.

Findings

Conditions for global existence of solutions.

Criteria for finite-time blow-up.

Analysis of nonlocal convolution effects.

Abstract

This paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is reflected by a convolution integral in the space variables. The Fourier transform of the convolution kernel is nonnegative and satisfies a certain growth condition at infinity. For initial data in $L^{2}$ Sobolev spaces, conditions for global existence or finite time blow-up of the solutions of the Cauchy problem are established.