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Global Attractor of Reaction–Diffusion Gene Regulatory Networks with S-Type Delay

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Abstract

This paper studies the existence of the global attractor for the reaction–diffusion gene regulatory networks with s-type delay. Firstly, two Hanalay inequalities are proposed and proved, then the uniform boundedness theorem is proved by using these inequalities and semigroup theory and functional analysis technique, the operator splitting technique based on the superposition principle is also utilized to deal with the asymptotic compact of the semigroup. Then some sufficient conditions on existence of global attractor are established, which are easily verifiable and have a wider application range. At last, two examples are discussed to validate the effectiveness of our results. Moreover, the simulation is given by using Matlab.

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Acknowledgements

We feel specially honored to meet the serious editor and rigorous reviewers. The paper is significantly improved after following their suggestion and comments. We are grateful to them. This work is founded by Shandong Provincial Natural Science Foundation under Contract No. ZR2017BA014, National Natural Science Foundation of China under Contract No. 61573008, and the Development Program for Defense Department of China under Contract No. C1520110002. The first author thanks Shandong University of Science and Technology for funding his visiting career in University of Southern California, Los Angeles, CA, from Jan. 2018–Jan. 2019.

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Authors and Affiliations

  1. College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, China

    Xiao Liang & Xianglai Zhuo

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100094, China

    Ruili Wang

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  1. Xiao Liang
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  2. Xianglai Zhuo
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  3. Ruili Wang
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Correspondence to Xiao Liang.

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Liang, X., Zhuo, X. & Wang, R. Global Attractor of Reaction–Diffusion Gene Regulatory Networks with S-Type Delay. Neural Process Lett 51, 1557–1577 (2020). https://doi.org/10.1007/s11063-019-10164-z

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