# American Institute of Mathematical Sciences

December  2019, 14(4): 789-804. doi: 10.3934/nhm.2019032

## Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays

 1 Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul 03722, Republic of Korea 2 Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università di L'Aquila, Via Vetoio, 67010 L'Aquila, Italy

* Corresponding author: Cristina Pignotti

Received  February 2019 Revised  June 2019 Published  October 2019

Fund Project: The first author was supported by National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation. The research of the second author was partially supported by the GNAMPA 2018 project Analisi e controllo di modelli differenziali non lineari (INdAM)

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals $N$ tends to infinity, the $N$-particle system can be well approximated by a delayed Vlasov alignment equation. Furthermore, we also establish the global existence of measure-valued solutions for the delayed Vlasov alignment equation and its large-time asymptotic behavior.

Citation: Young-Pil Choi, Cristina Pignotti. Emergent behavior of Cucker-Smale model with normalized weights and distributed time delays. Networks & Heterogeneous Media, 2019, 14 (4) : 789-804. doi: 10.3934/nhm.2019032
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