# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022036
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## Approaching constant steady states in a Keller-Segel-Stokes system with subquadratic logistic growth

 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731 China

Received  November 2021 Revised  January 2022 Early access March 2022

Fund Project: The first author is supported by the Applied Fundamental Research Program of Sichuan Province grant No. 2020YJ0264

In this paper, we investigate the large time behavior of the generalized solution to the Keller-Segel-Stokes system with logistic growth
 $\rho n-rn^{\alpha }$
in a bounded domain
 $\Omega\subset \mathbb R^d$
 $(d\in\{2, 3\})$
, as given by
 $\begin{equation*} \left\{ \begin{array}{l} &n_t+{{\bf{u}}}\cdot\nabla n = \Delta n-\chi\nabla\cdot\big(n\nabla c\big)+\rho n-rn^{\alpha }, \\ &c_t+{{\bf{u}}}\cdot\nabla c = \Delta c-c+n, \\ &{{\bf{u}}}_t+\nabla P = \Delta{{\bf{u}}}+n\nabla\phi, \\ &\nabla\cdot{{\bf{u}}} = 0 \end{array} \right. \end{equation*}$
for the unknown
 $(n, c, {{\bf{u}}}, P)$
, with prescribed and suitably smooth
 $\phi$
. Our result shows that if
 $\alpha$
,
 $\chi$
,
 $\rho$
and
 $r$
satisfy
 $\alpha > \frac{2d-2}{d}\quad\mathrm{and}\quad\chi^2< K\rho^{ \frac{\alpha -3}{\alpha -1}}r^{ \frac{2}{\alpha -1}}$
with some positive constant
 $K$
depending on
 $\alpha$
,
 $\Omega$
and
 $\phi$
, the generalized solution converges to a constant steady state ((
 $\frac{\rho}{r})^{ \frac{1}{\alpha -1}}, ( \frac{\rho}{r})^{ \frac{1}{\alpha -1}}, {\bf 0}$
) after a large time. Our proof is based on the decay property of a functional involving
 $n$
,
 $c$
and
 ${\bf{u}}$
.
Citation: Yu Tian. Approaching constant steady states in a Keller-Segel-Stokes system with subquadratic logistic growth. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022036
##### References:
 [1] N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4 (2017), 31-67.  doi: 10.1090/btran/17. [2] 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. [3] X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141. [4] X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. [5] C. Dellache and P. A. Meyer, Probabilities and Potential, Amsterdam-New York, 1978. [6] M. Ding and W. Lyu, Generalized solutions to a chemotaxis-fluid system with arbirtary superlinear degaradation, in print. [7] E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001. [8] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1998), 633-683. [9] K. Hideo, M. Masanari and S. Yoshie, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016. [10] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569. [11] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363. [12] B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.  doi: 10.1016/j.aml.2016.08.003. [13] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6. [14] H. Jin, Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity, Discrete Contin. Dyn. Syst., 38 (2018), 3595-3616.  doi: 10.3934/dcds.2018155. [15] H. Jin and T. Xiang, Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller-Segel model, C. R. Math. Acad. Sci., 356 (2018), 875-885.  doi: 10.1016/j.crma.2018.07.002. [16] L. Johannes, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [17] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017. [18] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Bio., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [19] A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.  doi: 10.4310/CMS.2012.v10.n2.a7. [20] N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505.  doi: 10.1007/s00526-012-0558-4. [21] N. Mizoguchi, Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system, J. Math. Pures Appl., 136 (2020), 203-238.  doi: 10.1016/j.matpur.2019.10.004. [22] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. [23] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042. [24] T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal., 30 (1997), 3837-3842.  doi: 10.1016/S0362-546X(96)00256-8. [25] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [26] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [27] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations., Funkcial. Ekvac., 44 (2001), 441-469. [28] 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. [29] 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. [30] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. [31] Y. Tao and M. Winkler, Taxis-driven formation of singular hotspots in a May-Nowak type model for virus infection, SIAM J. Math. Anal., 53 (2021), 1411-1433.  doi: 10.1137/20M1362851. [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [33] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102. [34] G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069. [35] Y. Wang, M. Winkler and Z. Xiang, Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation, Adv. Nonlinear Anal., 10 (2021), 707-731.  doi: 10.1515/anona-2020-0158. [36] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027. [37] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010. [38] 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. [39] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865. [40] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2. [41] 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. [42] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [43] M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009. [44] 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. [45] M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. [46] M. Winkler, Attractiveness of constant states in logistic-type Keller-Segel systems involving subquadratic growth restrictions, Adv. Nonlinear Stud., 20 (2020), 795-817.  doi: 10.1515/ans-2020-2107. [47] M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, preprint, arXiv: 10.2422/2036. [48] M. Winkler, Small-mass solutions in the two-dimensional Keller-Segel system coupled to the Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 2041-2080.  doi: 10.1137/19M1264199. [49] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [50] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071. [51] T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.  doi: 10.1016/j.jmaa.2017.11.022. [52] L. Xie and T. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal. Real World Appl., 37 (2017), 14-30.  doi: 10.1016/j.nonrwa.2017.02.005. [53] C. Yang C, X. Cao, Z. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093. [54] H. Yu, W. Wang and S. Zheng, Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.  doi: 10.1016/j.jmaa.2017.12.048. [55] W. Zhang, P. Niu and S. Liu, Large time behavior in a chemotaxis model with logistic growth and indirect signal production, Nonlinear Anal. Real World Appl., 50 (2019), 484-497.  doi: 10.1016/j.nonrwa.2019.05.002.

show all references

##### References:
 [1] N. Bellomo and M. Winkler, Finite-time blow-up in a degenerate chemotaxis system with flux limitation, Trans. Amer. Math. Soc. Ser. B, 4 (2017), 31-67.  doi: 10.1090/btran/17. [2] 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. [3] X. Cao, Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3369-3378.  doi: 10.3934/dcdsb.2017141. [4] X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. [5] C. Dellache and P. A. Meyer, Probabilities and Potential, Amsterdam-New York, 1978. [6] M. Ding and W. Lyu, Generalized solutions to a chemotaxis-fluid system with arbirtary superlinear degaradation, in print. [7] E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.  doi: 10.1016/j.nonrwa.2014.07.001. [8] M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1998), 633-683. [9] K. Hideo, M. Masanari and S. Yoshie, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal., 270 (2016), 1663-1683.  doi: 10.1016/j.jfa.2015.10.016. [10] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.  doi: 10.1002/mma.569. [11] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.  doi: 10.1017/S0956792501004363. [12] B. Hu and Y. Tao, Boundedness in a parabolic-elliptic chemotaxis-growth system under a critical parameter condition, Appl. Math. Lett., 64 (2017), 1-7.  doi: 10.1016/j.aml.2016.08.003. [13] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6. [14] H. Jin, Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity, Discrete Contin. Dyn. Syst., 38 (2018), 3595-3616.  doi: 10.3934/dcds.2018155. [15] H. Jin and T. Xiang, Chemotaxis effect vs. logistic damping on boundedness in the 2-D minimal Keller-Segel model, C. R. Math. Acad. Sci., 356 (2018), 875-885.  doi: 10.1016/j.crma.2018.07.002. [16] L. Johannes, Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.  doi: 10.3934/dcdsb.2015.20.1499. [17] K. Kang and A. Stevens, Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.  doi: 10.1016/j.na.2016.01.017. [18] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theo. Bio., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5. [19] A. Lorz, A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.  doi: 10.4310/CMS.2012.v10.n2.a7. [20] N. Mizoguchi, Global existence for the Cauchy problem of the parabolic-parabolic Keller-Segel system on the plane, Calc. Var. Partial Differential Equations, 48 (2013), 491-505.  doi: 10.1007/s00526-012-0558-4. [21] N. Mizoguchi, Finite-time blowup in Cauchy problem of parabolic-parabolic chemotaxis system, J. Math. Pures Appl., 136 (2020), 203-238.  doi: 10.1016/j.matpur.2019.10.004. [22] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. [23] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.  doi: 10.1155/S1025583401000042. [24] T. Nagai and T. Senba, Behavior of radially symmetric solutions of a system related to chemotaxis, Nonlinear Anal., 30 (1997), 3837-3842.  doi: 10.1016/S0362-546X(96)00256-8. [25] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [26] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A., 51 (2002), 119-144.  doi: 10.1016/S0362-546X(01)00815-X. [27] K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations., Funkcial. Ekvac., 44 (2001), 441-469. [28] 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. [29] 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. [30] Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1. [31] Y. Tao and M. Winkler, Taxis-driven formation of singular hotspots in a May-Nowak type model for virus infection, SIAM J. Math. Anal., 53 (2021), 1411-1433.  doi: 10.1137/20M1362851. [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003. [33] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences, 102 (2005), 2277-2282.  doi: 10.1073/pnas.0406724102. [34] G. Viglialoro, Very weak global solutions to a parabolic-parabolic chemotaxis-system with logistic source, J. Math. Anal. Appl., 439 (2016), 197-212.  doi: 10.1016/j.jmaa.2016.02.069. [35] Y. Wang, M. Winkler and Z. Xiang, Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation, Adv. Nonlinear Anal., 10 (2021), 707-731.  doi: 10.1515/anona-2020-0158. [36] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.  doi: 10.1016/j.jde.2015.08.027. [37] Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differential Equations, 261 (2016), 4944-4973.  doi: 10.1016/j.jde.2016.07.010. [38] 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. [39] M. Winkler, Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.  doi: 10.1080/03605302.2011.591865. [40] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.  doi: 10.1007/s00526-015-0922-2. [41] 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. [42] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.  doi: 10.1016/j.jde.2010.02.008. [43] M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: global weak solutions and asymptotic stabilization, J. Funct. Anal., 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009. [44] 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. [45] M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), Paper No. 69, 40 pp. doi: 10.1007/s00033-018-0935-8. [46] M. Winkler, Attractiveness of constant states in logistic-type Keller-Segel systems involving subquadratic growth restrictions, Adv. Nonlinear Stud., 20 (2020), 795-817.  doi: 10.1515/ans-2020-2107. [47] M. Winkler, $L^1$ solutions to parabolic Keller-Segel systems involving arbitrary superlinear degradation, preprint, arXiv: 10.2422/2036. [48] M. Winkler, Small-mass solutions in the two-dimensional Keller-Segel system coupled to the Navier-Stokes equations, SIAM J. Math. Anal., 52 (2020), 2041-2080.  doi: 10.1137/19M1264199. [49] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.  doi: 10.1007/s00332-014-9205-x. [50] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729.  doi: 10.1016/j.jmaa.2008.07.071. [51] T. Xiang, How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.  doi: 10.1016/j.jmaa.2017.11.022. [52] L. Xie and T. Xiao, Global existence and boundedness in a 2D Keller-Segel-Stokes system, Nonlinear Anal. Real World Appl., 37 (2017), 14-30.  doi: 10.1016/j.nonrwa.2017.02.005. [53] C. Yang C, X. Cao, Z. Jiang and S. Zheng, Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.  doi: 10.1016/j.jmaa.2015.04.093. [54] H. Yu, W. Wang and S. Zheng, Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.  doi: 10.1016/j.jmaa.2017.12.048. [55] W. Zhang, P. Niu and S. Liu, Large time behavior in a chemotaxis model with logistic growth and indirect signal production, Nonlinear Anal. Real World Appl., 50 (2019), 484-497.  doi: 10.1016/j.nonrwa.2019.05.002.
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