Electron. J. Differential Equations, Vol. 2017 (2017), No. 55, pp. 1-7.

Global well-posedness for nonlinear nonlocal Cauchy problems arising in elasticity

Hantaek Bae, Suleyman Ulusoy

Abstract:
In this article, we prove global well-posedness for a family of one dimensional nonlinear nonlocal Cauchy problems arising in elasticity. We consider the equation
$$ u_{tt}-\delta Lu_{xx}=\big(\beta \ast [(1-\delta)u+u^{2n+1}]\big)_{xx}\,, $$
where $L$ is a differential operator, $\beta$ is an integral operator, and $\delta =0$ or 1. (Here, the case $\delta=1$ represents the additional doubly dispersive effect.) We prove the global well-posedness of the equation in energy spaces.

Submitted May 24, 2016. Published February 22, 2017.
Math Subject Classifications: 35Q74, 35L15, 74B20.
Key Words: Nonlinear nonlocal wave equations; kernel function; global solution.

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Hantaek Bae
Department of Mathematical Sciences
Ulsan National Institute of Science and Technology
(UNIST), Korea
email: hantaek@unist.ac.kr
Süleyman Ulusoy
Department Of Mathematics and Natural Sciences
American University of Ras al Khaimah
PO Box 10021,Ras Al Khaimah, UAE
email: suleyman.ulusoy@aurak.ac.ae, suleymanulusoy@yahoo.com

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