A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity

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June 2022, 27(6): 3487-3513. doi: 10.3934/dcdsb.2021193

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 Published June 2022 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) $

.

Keywords: Chemotaxis, asymptotic behavior, boundedness, logistic source, convergence rate.

Mathematics Subject Classification: 92C17, 35B40, 35K55, 35K57, 42A15.

Citation: Jie Zhao. A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3487-3513. doi: 10.3934/dcdsb.2021193

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.
[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.
[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.
[4]	M. Ding, W. 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.
[5]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[6]	K. Fujie, M. 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.
[7]	K. Fujie, M. 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.
[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.
[9]	E. Galakhov, O. 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.
[10]	D. D. Haroske, H. D. Triebel, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.
[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.
[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.
[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]	S. Ishida, K. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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, Funkc. Ekvacioj, 40 (1997), 411-433.
[26]	T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28.
[27]	L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.
[28]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	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.
[39]	L. Wang, C. 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.
[40]	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. Ser. B, 25 (2020), 191-221. doi: 10.3934/dcdsb.2019178.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[49]	J. Zhao, C. Mu, L. 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.
[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.
[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.

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.
[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.
[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.
[4]	M. Ding, W. 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.
[5]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[6]	K. Fujie, M. 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.
[7]	K. Fujie, M. 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.
[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.
[9]	E. Galakhov, O. 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.
[10]	D. D. Haroske, H. D. Triebel, Sobolev Spaces, Elliptic Equations, European Mathematical Society, Zurich, 2008.
[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.
[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.
[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]	S. Ishida, K. 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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, Funkc. Ekvacioj, 40 (1997), 411-433.
[26]	T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28.
[27]	L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733-737.
[28]	K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[38]	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.
[39]	L. Wang, C. 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.
[40]	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. Ser. B, 25 (2020), 191-221. doi: 10.3934/dcdsb.2019178.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[49]	J. Zhao, C. Mu, L. 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.
[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.
[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.

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