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

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.

15 pages. "Section 6 The Anisotropic Case" added and minor changes. Accepted for publication in Nonlinearity

References in corpus (1)

• Blow-up and global existence for a general class of nonlocal nonlinear coupled wave equations

Cited by in corpus (5)

• The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model
• Existence and Stability of Traveling Waves for a Class of Nonlocal Nonlinear Equations
• On the convergence of the nonlocal nonlinear model to the classical elasticity equation
• On the full dispersion Kadomtsev-Petviashvili equations for dispersive elastic waves
• Global existence and blow-up of solutions for a general class of doubly dispersive nonlocal nonlinear wave equations