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November  2021, 29(5): 3261-3279. doi: 10.3934/era.2021037

## Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

 Faculty of Education, Sichuan Vocational and Technical College, Suining 629000, China

* Corresponding author: congnv333@163.com

Received  February 2021 Revised  April 2021 Published  November 2021 Early access  May 2021

Fund Project: This work is supported by Science Research Fund of Education Department of Sichuan Province of China under grant 18ZB0537

In this work, the fully parabolic chemotaxis-competition system with loop
 $\begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*}$
is considered under the homogeneous Neumann boundary condition, where
 $x\in\Omega, t>0$
,
 $\Omega\subset \mathbb{R}^{n} (n\leq 3)$
is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters
 $\mu_1, \mu_2$
are sufficiently large, then the system possesses a unique and global classical solution for
 $n\leq 3$
. Specifically, when
 $n = 2$
, the global boundedness can be attained without any constraints on
 $\mu_1, \mu_2$
.
Citation: Chun Huang. Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop. Electronic Research Archive, 2021, 29 (5) : 3261-3279. doi: 10.3934/era.2021037
##### References:
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show all references

##### References:
 [1] X. Bai and M. Winkler, Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.  doi: 10.1512/iumj.2016.65.5776.  Google Scholar [2] N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.  Google Scholar [3] T. Black, Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1253-1272.  doi: 10.3934/dcdsb.2017061.  Google Scholar [4] E. Espejo, K. Vilches and C. Conca, A simultaneous blow-up problem arising in tumor Modeling, J. Math. Biol., 79 (2019), 1357-1399.  doi: 10.1007/s00285-019-01397-6.  Google Scholar [5] H. Kn$\acute{u}$tsd$\acute{o}$ttir, E. P$\acute{a}$lsson and L. Edelstein-Keshet, Mathematical model of macrophage-facilitated breast cancer cells invasion, J. Theor. Biol., 357 (2014), 184-199.  doi: 10.1016/j.jtbi.2014.04.031.  Google Scholar [6] X. Li and Y. Wang, Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2717-2729.  doi: 10.3934/dcdsb.2017132.  Google Scholar [7] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar'e Anal. Non Lin'eaire, 31 (2014), 851-875.  doi: 10.1016/j.anihpc.2013.07.007.  Google Scholar [8] X. Pan, L. Wang, J. Zhang and J. Wang, Boundedness in a three-dimensional two-species chemotaxis system with two chemicals, Z. Angew. Math. Phys., 71 (2020). doi: 10.1007/s00033-020-1248-2.  Google Scholar [9] C. Stinner, C. Surulescu and M. Winkler, Global weak solutions in a PDE-ODE system modeling multiscale cancer cell invasion, SIAM J. Math. Anal., 46 (2014), 1969-2007.  doi: 10.1137/13094058X.  Google Scholar [10] Y. Tao and M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.  doi: 10.1016/j.jde.2014.04.014.  Google Scholar [11] Y. Tao and M. Winkler, Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.  doi: 10.3934/dcdsb.2015.20.3165.  Google Scholar [12] X. Tu, C. Mu, P. Zheng and K. Lin, Global dynamics in a two species chemotaxis competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.  doi: 10.3934/dcds.2018156.  Google Scholar [13] X. Tu, C. Mu and S. Qiu, Global asymptotic stability in a parabolic-elliptic chemotaxis system with competitive kinetics and loop, Appl. Anal., (2020). doi: 10.1080/00036811.2020.1783536.  Google Scholar [14] X. Tu, C. Mu and S. Qiu, Boundedness and convergence of constant equilibria in a two-species chemotaxis-competition system with loop, Nonlinear Anal., 198 (2020), 111923. doi: 10.1016/j.na.2020.111923.  Google Scholar [15] X. Tu, C. Mu, S. Qiu and L. Yang, Boundedness in the higher-dimensional fully parabolic chemotaxis-competition system with loop, Z. Angew. Math. Phys., 71 (2020), 185. doi: 10.1007/s00033-020-01413-6.  Google Scholar [16] X. Tu, C.-L. Tang and S. Qiu, The phenomenon of large population densities in a chemotaxis-competition system with loop, J. Evol. Equ., (2020). doi: 10.1007/s00028-020-00650-6.  Google Scholar [17] L. Wang, J. Zhang C. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two Chemicals, Discrete Contin. Dyn. Syst. B, 25 (2020), 191-221. doi: 10.3934/dcdsb.2019178.  Google Scholar [18] L. Wang and C. Mu, A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B, 25 (2020), 4585-4601. doi: 10.3934/dcdsb.2020114.  Google Scholar [19] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.  Google Scholar [20] H. Yu, W. Wang and S. Zheng, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514. doi: 10.1088/1361-6544/aa96c9.  Google Scholar [21] Q. Zhang, Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32. doi: 10.1016/j.aml.2018.03.012.  Google Scholar
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