Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime

Jan Haskovec; Ioannis Markou

doi:10.3934/krm.2020027

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2020, Volume 13, Issue 4: 795-813. Doi: 10.3934/krm.2020027

This issue Previous Article Slow flocking dynamics of the Cucker-Smale ensemble with a chemotactic movement in a temperature field Next Article Well-posedness for boundary value problems for coagulation-fragmentation equations

Asymptotic flocking in the Cucker-Smale model with reaction-type delays in the non-oscillatory regime

Jan Haskovec^1, , and
Ioannis Markou^2,

1.
Computer, Electrical and Mathematical Sciences & Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
2.
Institute of Applied and Computational Mathematics (IACM-FORTH), N. Plastira 100, Vassilika Vouton, GR - 700 13 Heraklion, Crete, Greece

^* Corresponding author: Jan Haskovec
^* Corresponding author: Jan Haskovec

Received: August 31, 2019

Revised: January 31, 2020

Published: May 2020

The first author is supported by KAUST baseline funds

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Abstract

We study a variant of the Cucker-Smale system with reaction-type delay. Using novel backward-forward and stability estimates on appropriate quantities we derive sufficient conditions for asymptotic flocking of the solutions. These conditions, although not explicit, relate the velocity fluctuation of the initial datum and the length of the delay. If satisfied, they guarantee monotone decay (i.e., non-oscillatory regime) of the velocity fluctuations towards zero for large times. For the simplified setting with only two agents and constant communication rate the Cucker-Smale system reduces to the delay negative feedback equation. We demonstrate that in this case our method provides the sharp condition for the size of the delay such that the solution be non-oscillatory. Moreover, we comment on the mathematical issues appearing in the formal macroscopic description of the reaction-type delay system.

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Mathematics Subject Classification: Primary: 34K05, 82C22; Secondary: 34D05, 92D50.

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