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

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

2010-09-07

arxiv.org/abs/1009.1266v2

Mathematics

Analysis of PDEs

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

Scientific paper

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.

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