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

Albert Erkip; Abba I. Ramadan

doi:10.3934/cpaa.2017105

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2017, Volume 16, Issue 6: 2125-2132. Doi: 10.3934/cpaa.2017105

This issue Previous Article Multiple solutions for a fractional nonlinear Schrödinger equation with local potential Next Article A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity

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

Albert Erkip^1, , and
Abba I. Ramadan^1,2,

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
* Corresponding author

Received: November 30, 2016

Revised: January 31, 2017

Published: September 2017

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 74H20, 74J30; Secondary: 35Q51, 35A15.

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