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doi: 10.3934/dcdsb.2021193
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A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity

College of Mathematics and Information, China West Normal University, NanChong 637000, China

* Corresponding author: Jie Zhao

Received  November 2020 Revised  June 2021 Early access July 2021

This paper deals with the dynamical properties of the quasilinear parabolic-parabolic chemotaxis system
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(\frac{u}{v} \nabla v)+\mu u- \mu u^{2}, \, \, \, &x\in\Omega, \, \, \, t>0, \\ v_{t} = \Delta v-v+u, &x\in\Omega, \, \, \, t>0, \end{array} \right. \end{eqnarray*} $
under homogeneous Neumann boundary conditions in a convex bounded domain
$ \Omega\subset\mathbb{R}^{n} $
,
$ n\geq2 $
, with smooth boundary.
$ \chi>0 $
and
$ \mu>0 $
,
$ D(u) $
is supposed to satisfy the behind properties
$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $
It is shown that there is a positive constant
$ m_{*} $
such that
$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $
for all
$ t\geq0 $
. Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution
$ (1, 1) $
.
Citation: Jie Zhao. A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021193
References:
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M. DingW. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. RWA, 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.  Google Scholar

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K. FujieM. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149.  Google Scholar

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H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr, 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

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Z. Jia and Z. Yang, Global boundedness to a parabolic-parabolic chemotaxis model with nonlinear diffusion and singular sensitivity, J. Math. Anal. Appl., 475 (2019), 139-153.  doi: 10.1016/j.jmaa.2019.02.022.  Google Scholar

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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

[18]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[19]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[20]

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

[21]

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

[22]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433.   Google Scholar

[26]

T. NagaiT. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28.   Google Scholar

[27]

L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.   Google Scholar

[28]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[29]

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

[30]

G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differential Equations, 268 (2020), 4320-4373.  doi: 10.1016/j.jde.2019.10.027.  Google Scholar

[31]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differential Equations, 269 (2020), 1484-1520.  doi: 10.1016/j.jde.2020.01.008.  Google Scholar

[32]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction- repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689. doi: 10.1142/S0218202520500517.  Google Scholar

[33]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[35]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.  Google Scholar

[36]

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.  Google Scholar

[37]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[38]

L. WangC. MuX. 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

[39]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.  Google Scholar

[40]

L. WangJ. ZhangC. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.  Google Scholar

[41]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[42]

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.  Google Scholar

[43]

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

[44]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[45]

M. Winkler, Global asymptotic stability of constant equilibria ina 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

[46]

M. Winkler, Aggregation versus 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.  Google Scholar

[47]

J. Zhao, Large time behavior of solution to quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1737-1755.  doi: 10.3934/dcds.2020091.  Google Scholar

[48]

J. Zhao, Convergence rate of a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Math. Anal. Appl., 478 (2019), 625-633.  doi: 10.1016/j.jmaa.2019.05.047.  Google Scholar

[49]

J. ZhaoC. MuL. Wang and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., 99 (2020), 86-102.  doi: 10.1080/00036811.2018.1489955.  Google Scholar

[50]

X. Zhao, S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 2, 13 pp. doi: 10.1007/s00033-016-0749-5.  Google Scholar

[51]

X. Zhao and S. Zheng, Global existence and boundedness of solutions to achemotaxis system with singular sensitivity and logistic-type source, J. Differential Equations, 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.  Google Scholar

show all references

References:
[1]

P. Biler and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitational interaction of particles. I, Colloq. Mathematicum, 66 (1994), 319-334.  doi: 10.4064/cm-66-2-319-334.  Google Scholar

[2]

T. Cieálak 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.  Google Scholar

[3]

T. Cieálak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076.  doi: 10.1088/0951-7715/21/5/009.  Google Scholar

[4]

M. DingW. Wang and S. Zhou, Global existence of solutions to a fully parabolic chemotaxis system with singular sensitivity and logistic source, Nonlinear Anal. RWA, 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.  Google Scholar

[5]

A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar

[6]

K. FujieM. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic Keller-Segel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224.  doi: 10.1002/mma.3149.  Google Scholar

[7]

K. FujieM. Winkler and T. Yokota, Blow-up prevention by logistic sources in a parabolic-elliptic Keller-Segel system with singular sensitivity, Nonlinear Anal, 109 (2014), 56-71.  doi: 10.1016/j.na.2014.06.017.  Google Scholar

[8]

H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr, 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.  Google Scholar

[9]

E. GalakhovO. Salieva and J. I. Tello, On a parabolic-elliptic system with chemotaxis and logistic type growth, J. Differential Equations, 261 (2016), 4631-4647.  doi: 10.1016/j.jde.2016.07.008.  Google Scholar

[10]

D. D. Haroske, H. D. Triebel, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.  Google Scholar

[11]

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.  Google Scholar

[12]

M. A. Herrero and J. J. L. Velsazquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore Pisa Cl. Sci., 24 (1997), 633-683.   Google Scholar

[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.  Google Scholar

[14]

S. IshidaK. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010.  doi: 10.1016/j.jde.2014.01.028.  Google Scholar

[15]

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.  Google Scholar

[16]

Z. Jia and Z. Yang, Global boundedness to a parabolic-parabolic chemotaxis model with nonlinear diffusion and singular sensitivity, J. Math. Anal. Appl., 475 (2019), 139-153.  doi: 10.1016/j.jmaa.2019.02.022.  Google Scholar

[17]

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

[18]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theoret. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[19]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[20]

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

[21]

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

[22]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, 40 (1997), 411-433.   Google Scholar

[26]

T. NagaiT. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28.   Google Scholar

[27]

L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.   Google Scholar

[28]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.   Google Scholar

[29]

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

[30]

G. Ren and B. Liu, Global dynamics for an attraction-repulsion chemotaxis model with logistic source, J. Differential Equations, 268 (2020), 4320-4373.  doi: 10.1016/j.jde.2019.10.027.  Google Scholar

[31]

G. Ren and B. Liu, Global existence and asymptotic behavior in a two-species chemotaxis system with logistic source, J. Differential Equations, 269 (2020), 1484-1520.  doi: 10.1016/j.jde.2020.01.008.  Google Scholar

[32]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a quasilinear attraction- repulsion chemotaxis model with nonlinear signal production and logistic-type source, Math. Models Methods Appl. Sci., 30 (2020), 2619-2689. doi: 10.1142/S0218202520500517.  Google Scholar

[33]

Y. Tao and Z.-A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 1 (2013), 1-36.  doi: 10.1142/S0218202512500443.  Google Scholar

[34]

Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subscritical sensitivity, J. Differential Equations, 252 (2012), 692-715.  doi: 10.1016/j.jde.2011.08.019.  Google Scholar

[35]

Y. Tao and M. Winkler, Large time behavior in a multi-dimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.  doi: 10.1137/15M1014115.  Google Scholar

[36]

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.  Google Scholar

[37]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.  doi: 10.1080/03605300701319003.  Google Scholar

[38]

L. WangC. MuX. 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

[39]

L. WangC. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Differential Equations, 256 (2014), 1847-1872.  doi: 10.1016/j.jde.2013.12.007.  Google Scholar

[40]

L. WangJ. ZhangC. Mu and X. Hu, Boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 191-221.  doi: 10.3934/dcdsb.2019178.  Google Scholar

[41]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.  doi: 10.1002/mma.1146.  Google Scholar

[42]

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.  Google Scholar

[43]

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

[44]

M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.  doi: 10.1016/j.jmaa.2011.05.057.  Google Scholar

[45]

M. Winkler, Global asymptotic stability of constant equilibria ina 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

[46]

M. Winkler, Aggregation versus 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.  Google Scholar

[47]

J. Zhao, Large time behavior of solution to quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1737-1755.  doi: 10.3934/dcds.2020091.  Google Scholar

[48]

J. Zhao, Convergence rate of a quasilinear parabolic-elliptic chemotaxis system with logistic source, J. Math. Anal. Appl., 478 (2019), 625-633.  doi: 10.1016/j.jmaa.2019.05.047.  Google Scholar

[49]

J. ZhaoC. MuL. Wang and K. Lin, A quasilinear parabolic-elliptic chemotaxis-growth system with nonlinear secretion, Appl. Anal., 99 (2020), 86-102.  doi: 10.1080/00036811.2018.1489955.  Google Scholar

[50]

X. Zhao, S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), Paper No. 2, 13 pp. doi: 10.1007/s00033-016-0749-5.  Google Scholar

[51]

X. Zhao and S. Zheng, Global existence and boundedness of solutions to achemotaxis system with singular sensitivity and logistic-type source, J. Differential Equations, 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.  Google Scholar

[1]

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

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