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November  2017, 16(6): 2125-2132. doi: 10.3934/cpaa.2017105

## Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels

 1 Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla 34956, Istanbul, Turkey 2 Department of Mathematics, 405 Snow Hall, University of Kansas, Lawrence, KS 66045

* Corresponding author

Received  December 2016 Revised  February 2017 Published  July 2017

Fund Project: The second author is supported by TUBITAK Grant 2215 -Graduate Scholarship Programme for International Students.

In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: $~u_{tt}-a^{2}u_{xx}=(β * u^{p})_{xx}$, $~p>1$. Members of the class arise as mathematical models for the propagation of waves in a wide variety of situations. We assume that the kernel $β$ is a bell-shaped function satisfying some mild differentiability and growth conditions. Taking advantage of growth properties of bell-shaped functions, we give a simple proof for the existence of bell-shaped traveling wave solutions.

Citation: Albert Erkip, Abba I. Ramadan. Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2125-2132. doi: 10.3934/cpaa.2017105
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