Existence of traveling wave solutions to a nonlocal scalar equation with sign-changing kernel

doi:10.1016/j.jmaa.2020.124007

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Title:	Existence of traveling wave solutions to a nonlocal scalar equation with sign-changing kernel
Authors:	Ei, Shin-Ichiro;Guo, Jong-Shenq;Ishii, Hiroshi;Wu, Chin-Chin
Keywords:	Traveling wave;Wave speed;Nonlocal equation;Sign-changing kernel
Date:	2020-07-15
Issue Date:	2020-06-04 12:10:20 (UTC+8)
Publisher:	Academic Press
Abstract:	In this paper, we study the existence of traveling wave solutions connecting two constant states to a nonlocal scalar equation with sign-changing kernel. A typical example of such kernel in the neural fields is the Mexican hat type function. We first introduce a new notion of upper-lower-solution for the equation of wave profile for a given wave speed. Then, with the help of Schauder's fixed point theorem, we construct two different pairs of upper-lower-solutions to obtain traveling waves for a continuum of wave speeds under two different assumptions. Due to the sign-changing nature of the kernel, the wave profiles may take both positive and negative values. Finally, we analyze the limit of the right-hand tail of wave profiles. Under some further condition on the wave speeds, we prove that the right-hand tail limit of the wave profile does exist. In particular, we obtain the existence of nonnegative traveling waves connecting the unstable state 0 and the stable state 1 for wave speeds large enough.
Relation:	Journal of Mathematical Analysis and Applications 487(2), 124007
DOI:	10.1016/j.jmaa.2020.124007
Appears in Collections:	[Graduate Institute & Department of Mathematics] Journal Article

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