Traveling wave solutions for an integrodifference equation of higher order

Fuzhen Wu; Fuzhen Wu

doi:10.3934/math.2022902

AIMS Mathematics

2022, Volume 7, Issue 9: 16482-16497. doi: 10.3934/math.2022902

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Research article

Traveling wave solutions for an integrodifference equation of higher order

Fuzhen Wu ^,

Digital Information Technology School, Zhejiang Technical Institute of Economics, Hangzhou, Zhejiang 310018, China

Received: 13 May 2022 Revised: 29 June 2022 Accepted: 04 July 2022 Published: 08 July 2022
MSC : 39A70, 47G10, 92D25

This article is concerned with the minimal wave speed of traveling wave solutions for an integrodifference equation of higher order. Besides the operator may be nonmonotone, the kernel functions may be not Lebesgue measurable and integrable such that the equation has lower regularity. By constructing a proper set of potential wave profiles, we obtain the existence of smooth traveling wave solutions when the wave speed is larger than a threshold. Here, the profile set is obtained by giving a pair of upper and lower solutions. When the wave speed is the threshold, the existence of nontrivial traveling wave solutions is proved by passing to a limit function. Moreover, we obtain the nonexistence of nontrivial traveling wave solutions when the wave speed is smaller than the threshold.
- nonmonotone equation,
- upper-lower solutions,
- fixed point theorem,
- weaker regularity,
- minimal wave speed
Citation: Fuzhen Wu. Traveling wave solutions for an integrodifference equation of higher order[J]. AIMS Mathematics, 2022, 7(9): 16482-16497. doi: 10.3934/math.2022902

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Abstract

This article is concerned with the minimal wave speed of traveling wave solutions for an integrodifference equation of higher order. Besides the operator may be nonmonotone, the kernel functions may be not Lebesgue measurable and integrable such that the equation has lower regularity. By constructing a proper set of potential wave profiles, we obtain the existence of smooth traveling wave solutions when the wave speed is larger than a threshold. Here, the profile set is obtained by giving a pair of upper and lower solutions. When the wave speed is the threshold, the existence of nontrivial traveling wave solutions is proved by passing to a limit function. Moreover, we obtain the nonexistence of nontrivial traveling wave solutions when the wave speed is smaller than the threshold.

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