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

H. A. Erbay; S. Erbay; A. Erkip

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arXiv · 1807.02822

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

H. A. Erbay·S. Erbay·A. Erkip

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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.

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H. A. Erbay, S. Erbay, A. Erkip. 2018-07-08. Long-time existence of solutions to nonlocal nonlinear bidirectional wave equations. https://arxiv.org/abs/1807.02822

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