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

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

doi:10.3934/dcds.2019119

Article Contents

2019, Volume 39, Issue 5: 2877-2891. Doi: 10.3934/dcds.2019119

This issue Previous Article Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity Next Article A BDF2-approach for the non-linear Fokker-Planck equation

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

1.
Department of Natural and Mathematical Sciences, Faculty of Engineering, Ozyegin University, Cekmekoy 34794, Istanbul, Turkey
2.
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey

* Corresponding author: H. A. Erbay
* Corresponding author: H. A. Erbay

Received: June 30, 2018

Published: January 2019

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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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Mathematics Subject Classification: Primary: 35A01, 35L15; Secondary: 35L70, 35Q74, 74B20.

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References

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