# American Institute of Mathematical Sciences

## Boundedness and stabilization in a two-species chemotaxis system with two chemicals

 1 Key Lab of Intelligent Analysis and Decision on Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Liangchen Wang

Received  December 2018 Revised  March 2019 Published  July 2019

This paper deals with the two-species chemotaxis system with two chemicals
 $\begin{eqnarray*} \left\{ \begin{array}{llll} u_t = d_1\Delta u-\nabla\cdot(u\chi_1(v)\nabla v)+\mu_1 u(1-u-a_1w),\quad &x\in \Omega,\quad t>0,\\ v_t = d_2\Delta v-\alpha v+f_1(w),\quad &x\in\Omega,\quad t>0,\\ w_t = d_3\Delta w-\nabla\cdot(w\chi_2(z)\nabla z)+\mu_2 w(1-w-a_2u),\quad &x\in \Omega,\quad t>0,\\ z_t = d_4\Delta z-\beta z+f_2(u),\quad &x\in\Omega,\quad t>0, \end{array} \right. \end{eqnarray*}$
under homogeneous Neumann boundary conditions in a bounded domain
 $\Omega\subset \mathbb{R}^n$
(
 $n\geq1$
), where the parameters
 $d_1,d_2,d_3,d_4>0$
,
 $\mu_1,\mu_2>0$
,
 $a_1,a_2>0$
and
 $\alpha, \beta>0$
. The chemotactic function
 $\chi_i$
(
 $i = 1,2$
) and the signal production function
 $f_i$
(
 $i = 1,2$
) are smooth. If
 $n = 2$
, it is shown that this system possesses a unique global bounded classical solution provided that
 $|\chi'_i|$
(
 $i = 1,2$
) are bounded. If
 $n\leq3$
, this system possesses a unique global bounded classical solution provided that
 $\mu_i$
(
 $i = 1,2$
) are sufficiently large. Specifically, we first obtain an explicit formula
 $\mu_{i0}>0$
such that this system has no blow-up whenever
 $\mu_i>\mu_{i0}$
.
Moreover, by constructing suitable energy functions, it is shown that:
 $\bullet$
If
 $a_1,a_2\in(0,1)$
and
 $\mu_1$
and
 $\mu_2$
are sufficiently large, then any global bounded solution exponentially converges to
 $\bigg(\frac{1-a_1}{1-a_1a_2},f_1(\frac{1-a_2}{1-a_1a_2})/\alpha,\frac{1-a_2}{1-a_1a_2},$
 $f_2(\frac{1-a_1}{1-a_1a_2})/\beta\bigg)$
as
 $t\rightarrow\infty$
;
 $\bullet$
If
 $a_1>1>a_2>0$
and
 $\mu_2$
is sufficiently large, then any global bounded solution exponentially converges to
 $(0,f_1(1)/\alpha,1,0)$
as
 $t\rightarrow\infty$
;
 $\bullet$
If
 $a_1 = 1>a_2>0$
and
 $\mu_2$
is sufficiently large, then any global bounded solution algebraically converges to
 $(0,f_1(1)/\alpha,1,0)$
as
 $t\rightarrow\infty$
.
Citation: Liangchen Wang, Jing Zhang, Chunlai Mu, Xuegang Hu. Boundedness and stabilization in a two-species chemotaxis system with two chemicals. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019178
##### 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, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar [4] 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 [5] M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6. doi: 10.1016/j.aml.2015.12.001. Google Scholar [6] M. Eisenbach, Chemotaxis, (Imperial College Press, London, 2004. Google Scholar [7] T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar [9] 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. Google Scholar [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [11] 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. Google Scholar [12] D. Li, C. Mu, K. Lin and L.Wang, Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), Art. 56, 25 pp. doi: 10.1007/s00033-017-0800-1. Google Scholar [13] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046. doi: 10.3934/dcds.2016018. Google Scholar [14] K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260. doi: 10.3934/dcdsb.2017094. Google Scholar [15] K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429. Google Scholar [16] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar [17] M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249. doi: 10.1002/mma.4607. Google Scholar [18] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar [19] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319. doi: 10.3934/dcdsb.2017097. Google Scholar [20] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. Google Scholar [21] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2014), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627. Google Scholar [22] E. Nakaguchi and K. Osaki, $L^p$-estimates of solutions to $n$-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcial. Ekvac., 59 (2016), 51-66. doi: 10.1619/fesi.59.51. Google Scholar [23] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044. Google Scholar [24] M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar [25] M. Negreanu and J. I. 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. Google Scholar [26] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8. Google Scholar [27] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar [28] H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 1569-1587. Google Scholar [29] 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 [30] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar [31] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443. Google Scholar [32] 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 [33] 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 [34] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. Google Scholar [35] J. I. 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. Google Scholar [36] 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 [37] L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401. doi: 10.1016/j.jde.2017.11.019. Google Scholar [38] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. Google Scholar [39] 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 [40] 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. Google Scholar [41] G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661. doi: 10.1017/S0956792501004843. Google Scholar [42] T. Xiang, Boundedness and global existence in the higher-dimensional parabolic-parabolic chemotaxis system with/without growth source, J. Differential Equations, 258 (2015), 4275-4323. doi: 10.1016/j.jde.2015.01.032. Google Scholar [43] L. Xie 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 [44] 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 [45] P. Zheng and C. Mu, Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177. doi: 10.1007/s10440-016-0083-0. Google Scholar [46] Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9pp. doi: 10.1063/1.5011725. Google Scholar [47] 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 [48] Q. Zhang and Y. Li, Global solutions in a high-dimensional two-species chemotaxis model with Lotka-Volterra competitive kinetics, J. Math. Anal. Appl., 467 (2018), 751-767. doi: 10.1016/j.jmaa.2018.07.037. Google Scholar [49] P. Zheng, C. Mu and Y. Mi, Global stability in a two-competing-species chemotaxis system with two chemicals, Diff. Integ. Equa., 31 (2018), 547-558. Google Scholar

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##### 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, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876. doi: 10.1093/imamat/hxw036. Google Scholar [4] 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 [5] M. A. J. Chaplain and J. I. Tello, On the stability of homogeneous steady states of a chemotaxis system with logistic growth term, Appl. Math. Lett., 57 (2016), 1-6. doi: 10.1016/j.aml.2015.12.001. Google Scholar [6] M. Eisenbach, Chemotaxis, (Imperial College Press, London, 2004. Google Scholar [7] T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. Google Scholar [8] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. Google Scholar [9] 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. Google Scholar [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [11] 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. Google Scholar [12] D. Li, C. Mu, K. Lin and L.Wang, Convergence rate estimates of a two-species chemotaxis system with two indirect signal production and logistic source in three dimensions, Z. Angew. Math. Phys., 68 (2017), Art. 56, 25 pp. doi: 10.1007/s00033-017-0800-1. Google Scholar [13] K. Lin and C. Mu, Global dynamics in a fully parabolic chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 36 (2016), 5025-5046. doi: 10.3934/dcds.2016018. Google Scholar [14] K. Lin and C. Mu, Convergence of global and bounded solutions of a two-species chemotaxis model with a logistic source, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2233-2260. doi: 10.3934/dcdsb.2017094. Google Scholar [15] K. Lin, C. Mu and L. Wang, Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096. doi: 10.1002/mma.3429. Google Scholar [16] N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875. doi: 10.1016/j.anihpc.2013.07.007. Google Scholar [17] M. Mizukami, Boundedness and stabilization in a two-species chemotaxis-competition system of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249. doi: 10.1002/mma.4607. Google Scholar [18] M. Mizukami and T. Yokota, Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669. doi: 10.1016/j.jde.2016.05.008. Google Scholar [19] M. Mizukami, Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319. doi: 10.3934/dcdsb.2017097. Google Scholar [20] M. Mizukami, Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 269-278. Google Scholar [21] E. Nakaguchi and K. Osaki, Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation, Discrete Contin. Dyn. Syst. Ser. B, 18 (2014), 2627-2646. doi: 10.3934/dcdsb.2013.18.2627. Google Scholar [22] E. Nakaguchi and K. Osaki, $L^p$-estimates of solutions to $n$-dimensional parabolic-parabolic system for chemotaxis with subquadratic degradation, Funkcial. Ekvac., 59 (2016), 51-66. doi: 10.1619/fesi.59.51. Google Scholar [23] E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044. Google Scholar [24] M. Negreanu and J. I. Tello, On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781. doi: 10.1137/140971853. Google Scholar [25] M. Negreanu and J. I. 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. Google Scholar [26] K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147. doi: 10.1007/s11538-009-9396-8. Google Scholar [27] M. M. Porzio and V. Vespri, Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178. doi: 10.1006/jdeq.1993.1045. Google Scholar [28] H. Qiu and S. Guo, Global existence and stability in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. Ser. B, 24 (2018), 1569-1587. Google Scholar [29] 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 [30] C. Stinner, J. I. Tello and M. Winkler, Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626. doi: 10.1007/s00285-013-0681-7. Google Scholar [31] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443. Google Scholar [32] 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 [33] 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 [34] Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y. Google Scholar [35] J. I. 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. Google Scholar [36] 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 [37] L. Wang, C. Mu, X. Hu and P. Zheng, Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant, J. Differential Equations, 264 (2018), 3369-3401. doi: 10.1016/j.jde.2017.11.019. Google Scholar [38] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838. Google Scholar [39] 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 [40] M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. 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