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doi: 10.3934/dcdsb.2022071
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## Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals

 College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450000, China

*Corresponding author

Received  December 2021 Revised  March 2022 Early access March 2022

Fund Project: The first author is supported by Natural Science Foundation of Henan (212300410301), Key Research Plan of Henan Province Colleges(20B110021), the National Natural Science Foundation of China (11971446) and Doctor Foundation of Zhengzhou University of Light Industry(2018BSJJ062, 2017BSJJ071, 2017BSJJ072)

This paper studies the quasilinear attraction-repulsion chemotaxis system of two-species with two chemicals $u_{t} = \nabla\cdot( D_1(u)\nabla u)-\nabla\cdot( \Phi_1(u)\nabla v)$, $0 = \Delta v-v+w^{\gamma_1}$, $w_{t} = \nabla\cdot( D_2(w)\nabla w)+\nabla\cdot( \Phi_2(w)\nabla z)$, $0 = \Delta z-z+u^{\gamma_2}$, subject to the homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$($N\geq2$) with smooth boundary, where $\gamma_i>0$, $D_i,\Phi_i\in C^2[0,+\infty)$, $D_i(s)\ge(s+1)^{p_i},\; \Phi_i(s)\ge0$ for $s\ge 0$, and $\Phi_i(s)\le\chi_i s^{q_i}$ for $s>s_0$ with $\chi_i>0$, $p_i,q_i\in\mathbb{R}$ $(i = 1,2)$, $s_0>1$. It is shown that if $\gamma_1<\frac{2}{N}$ (or $\gamma_2<\frac{4}{N}$ with $\gamma_2\le1$), the global boundedness of solutions are guaranteed by the self-diffusion dominance of $u$ (or $w$) with $p_1>q_1+\gamma_1-1-\frac{2}{N}$ (or $p_2>q_2+\gamma_2-1-\frac{4}{N}$); if $p_j\ge q_i+\gamma_i- 1-\frac{2}{N}$, $i,j = 1,2$ (i.e. the self-diffusion of $u$ and $w$ are dominant), then the solutions are globally bounded; in particular, different from the results of the single-species chemotaxis system, for the critical case $p_j = q_i+\gamma_i- 1-\frac{2}{N}$, the global boundedness of the solutions can be obtained.

Citation: Miaoqing Tian, Shujuan Wang, Xia Xiao. Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022071
##### References:
 [1] R. Adams and F. John, Sobolev Spaces, 2$^{nd}$ edition, Elsvier Ltd., 2003. [2] 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. [3] T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832–5851. doi: 10.1016/j.jde.2012.01.045. [4] T. Cieslak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.  doi: 10.1007/s10440-013-9832-5. [5] T. Cieslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004. [6] A. 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Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x. [13] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [14] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6. [15] O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, 1968. doi: 10.1090/mmono/023. [16] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [17] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [18] M. Negreanu and J. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009. [19] X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math$\acute{e}$matique, 359 (2021), 161-168.  doi: 10.5802/crmath.148. [20] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443. [21] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [22] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019. [23] 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. [24] J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [25] J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413. [26] 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. Ser. B, 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114. [27] 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. [28] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [29] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023. [30] M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e. [31] M. Winkler and K. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [32] Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.  doi: 10.1016/j.jmaa.2014.03.084. [33] Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311.

show all references

##### References:
 [1] R. Adams and F. John, Sobolev Spaces, 2$^{nd}$ edition, Elsvier Ltd., 2003. [2] 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. [3] T. Cieslak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832–5851. doi: 10.1016/j.jde.2012.01.045. [4] T. Cieslak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146.  doi: 10.1007/s10440-013-9832-5. [5] T. Cieslak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.  doi: 10.1016/j.jde.2014.12.004. [6] A. Friedman, Partial Diferential Equations, Dover Books on Mathematics Series, Dover Publications, Incorporated, 2008. [7] D. Gilbarg and N. Trudinger, Elliptic Partial Diferential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. [8] X. He and S. Zheng, Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source, J. Math. Anal. Appl., 436 (2016), 970-982.  doi: 10.1016/j.jmaa.2015.12.058. [9] M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683. [10] L. Hong, M. Tian and S. Zheng, An attraction-repulsion chemotaxis system with nonlinear productions, J. Math. Anal. Appl., 484 (2020), 123703, 8 pp. doi: 10.1016/j.jmaa.2019.123703. [11] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jber. DMV, 105 (2003), 103-165. [12] D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270.  doi: 10.1007/s00332-010-9082-x. [13] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.  doi: 10.1016/j.jde.2004.10.022. [14] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Am. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6. [15] O. Ladyzenskaja, V. Solonnikov and N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, AMS, Providence, 1968. doi: 10.1090/mmono/023. [16] J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differential Equations, 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016. [17] J. Lankeit, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [18] M. Negreanu and J. Tello, Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.  doi: 10.1016/j.jde.2014.11.009. [19] X. Pan and L. Wang, Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production, C. R. Math$\acute{e}$matique, 359 (2021), 161-168.  doi: 10.5802/crmath.148. [20] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.  doi: 10.1142/S0218202512500443. [21] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019. [22] Y. Tao and M. Winkler, Persistence of mass in a chemotaxis system with logistic source, J. Differential Equations, 259 (2015), 6142-6161.  doi: 10.1016/j.jde.2015.07.019. [23] 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. [24] J. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [25] J. Tello and M. Winkler, Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.  doi: 10.1088/0951-7715/25/5/1413. [26] 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. Ser. B, 25 (2020), 4585-4601.  doi: 10.3934/dcdsb.2020114. [27] 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. [28] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.  doi: 10.1016/j.matpur.2013.01.020. [29] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023. [30] M. Winkler, A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.  doi: 10.1088/1361-6544/aaaa0e. [31] M. Winkler and K. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA., 72 (2010), 1044-1064.  doi: 10.1016/j.na.2009.07.045. [32] Q. Zhang and Y. Li, Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.  doi: 10.1016/j.jmaa.2014.03.084. [33] Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, ZAMM Z. Angew. Math. Mech., 96 (2016), 570-584.  doi: 10.1002/zamm.201400311.
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