Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels
Erkip, Albert and Ramadan, Abba Ibrahim (2017) Existence of traveling waves for a class of nonlocal nonlinear equations with bell shaped kernels. Communications on Pure and Applied Analysis, 16 (6). 2125 -2132. ISSN 1534-0392 (Print) 1553-5258 (Online) Published Online First http://dx.doi.org/10.3934/cpaa.2017105
Official URL: http://dx.doi.org/10.3934/cpaa.2017105
In this article we are concerned with the existence of traveling wave solutions of a general class of nonlocal wave equations: utt - a^2uxx = (beta* 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 beta 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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