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July 2019 Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system
Kanako Noda, Koichi Osaki
Hiroshima Math. J. 49(2): 251-271 (July 2019). DOI: 10.32917/hmj/1564106547

Abstract

We study the global-in-time existence and the asymptotic behavior of solutions to a one-dimensional chemotaxis system presented by Deneubourg (Insectes Sociaux 24 (1977)). The system models the self-organized nest construction process of social insects. In the limit as a time-scale coefficient tends to 0, the Deneubourg model reduces to a parabolic-parabolic Keller-Segel system with linear degradation. We first show the global-in-time existence of solutions. We next define the dynamical system of solutions and construct the global attractor. In addition, under the assumption of a large resting rate of worker insects, we construct a Lyapunov functional for the unique homogeneous equilibrium, which indicates that the global attractor consists only of the equilibrium.

Citation

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Kanako Noda. Koichi Osaki. "Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system." Hiroshima Math. J. 49 (2) 251 - 271, July 2019. https://doi.org/10.32917/hmj/1564106547

Information

Received: 6 February 2018; Revised: 8 February 2019; Published: July 2019
First available in Project Euclid: 26 July 2019

zbMATH: 07120742
MathSciNet: MR3984994
Digital Object Identifier: 10.32917/hmj/1564106547

Subjects:
Primary: 35K57, 37B25
Secondary: 35A01, 35B41, 35B45

Rights: Copyright © 2019 Hiroshima University, Mathematics Program

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Vol.49 • No. 2 • July 2019
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