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August  2018, 38(8): 3875-3898. doi: 10.3934/dcds.2018168

Global weak solution and boundedness in a three-dimensional competing chemotaxis

 1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 3 College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  September 2017 Revised  February 2018 Published  May 2018

Fund Project: The second author is partially supported by NSFC (Grant No. 11771062 and 11571062), the Fundamental Research Funds for the Central Universities (Grant No. 10611CDJXZ238826) and the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007), and the third author is partially supported by the Fundamental Research Funds for the Central Universities (Grant No. JBK1801059) and Chongqing Scientific & Technological Talents Program (Grant No. KJXX2017006)

We consider an initial-boundary value problem for a parabolic-parabolic-elliptic attraction-repulsion chemotaxis model
 $\left\{ \begin{array}{l}u_t = Δ u-χ\nabla·(u\nabla v)+ξ\nabla·(u\nabla w),&x∈ Ω,&t>0,\\v_t = Δ v-β v+α u,&x∈Ω,&t>0,\\0 = Δ w-δ w+γ u,&x∈Ω,&t>0\\\end{array} \right.$
in a bounded domain
 $Ω\subset \mathbb{R}^3$
with positive parameters
 $χ, ξ, α, β, γ$
and
 $δ$
.
It is firstly proved that if the repulsion dominates in the sense that
 $ξγ>χα$
, then for any choice of sufficiently smooth initial data
 $(u_0, v_0)$
the corresponding initial-boundary value problem is shown to possess a globally defined weak solution. To the best of our knowledge, this situation provides the first result on global existence of the above system in the three-dimensional setting when
 $ξγ>χα$
, and extends the results in Lin et al. (2016) [19] and Jin and Xiang (2017) [14] to more general case.
Secondly, if the initial data is appropriately small or the repulsion is enough strong in the sense that
 $ξγ$
is suitable large as related to
 $χα$
, then the classical solutions to the above system are uniformly-in-time bounded.
Citation: Hua Zhong, Chunlai Mu, Ke Lin. Global weak solution and boundedness in a three-dimensional competing chemotaxis. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3875-3898. doi: 10.3934/dcds.2018168
References:

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References:
 [1] Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050 [2] Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180 [3] Hai-Yang Jin, Tian Xiang. Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3071-3085. doi: 10.3934/dcdsb.2017197 [4] Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031 [5] Shijie Shi, Zhengrong Liu, Hai-Yang Jin. Boundedness and large time behavior of an attraction-repulsion chemotaxis model with logistic source. Kinetic & Related Models, 2017, 10 (3) : 855-878. doi: 10.3934/krm.2017034 [6] Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020027 [7] Rachidi B. Salako. Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5945-5973. doi: 10.3934/dcds.2019260 [8] Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072 [9] Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597 [10] Johannes Lankeit, Yulan Wang. Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6099-6121. doi: 10.3934/dcds.2017262 [11] Ling Liu, Jiashan Zheng. Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3357-3377. doi: 10.3934/dcdsb.2018324 [12] Youshan Tao. Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2705-2722. doi: 10.3934/dcdsb.2013.18.2705 [13] Sachiko Ishida. Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3463-3482. doi: 10.3934/dcds.2015.35.3463 [14] Wei Wang, Yan Li, Hao Yu. Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3663-3669. doi: 10.3934/dcdsb.2017147 [15] Chunhua Jin. Boundedness and global solvability to a chemotaxis-haptotaxis model with slow and fast diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1675-1688. doi: 10.3934/dcdsb.2018069 [16] Mengyao Ding, Wei Wang. Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4665-4684. doi: 10.3934/dcdsb.2018328 [17] Marcel Freitag. Global existence and boundedness in a chemorepulsion system with superlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5943-5961. doi: 10.3934/dcds.2018258 [18] T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125 [19] Pan Zheng. Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2039-2056. doi: 10.3934/dcdsb.2016035 [20] Radek Erban, Hyung Ju Hwang. Global existence results for complex hyperbolic models of bacterial chemotaxis. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1239-1260. doi: 10.3934/dcdsb.2006.6.1239

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