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January  2022, 27(1): 343-360. doi: 10.3934/dcdsb.2021045

Global existence in a chemotaxis system with singular sensitivity and signal production

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2. 

Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China

3. 

School of Science, Hubei University of Technology, Wuhan 430068, Hubei, China

* Corresponding author: Heping Ma

Received  October 2020 Revised  December 2020 Published  January 2022 Early access  February 2021

Fund Project: Guoqiang Ren is supported by NNSF of China(Grant No 12001214) and China Postdoctoral Science Foundation (Grant Nos. 2020M672319, 2020TQ0111). Heping Ma is supported by NNSF of China(Grant No 11801154)

In this work we consider the chemotaxis system with singular sensitivity and signal production in a two dimensional bounded domain. We present the global existence of weak solutions under appropriate regularity assumptions on the initial data. Our results generalize some well-known results in the literature.

Citation: Guoqiang Ren, Heping Ma. Global existence in a chemotaxis system with singular sensitivity and signal production. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 343-360. doi: 10.3934/dcdsb.2021045
References:
[1]

J. AhnK. Kang and J. Lee, Eventual smoothness and stabilization of global weak solutions in parabolic-elliptic chemotaxis systems with logarithmic sensitivity, Nonlinear Anal: Real World Appl., 49 (2019), 312-330.  doi: 10.1016/j.nonrwa.2019.03.012.

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[4]

T. Black, Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 119-137.  doi: 10.3934/dcdss.2020007.

[5]

L. Chen, F. Kong and Q. Wang, Global and exponential attractor of the repulsive Keller-Segel model with logarithmic sensitivity, Euro. J. Appl. Math., (2020), 1–19. doi: 10.1017/S0956792520000194.

[6]

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

[7]

M. Ding and X. Zhao, Global existence, boundedness and asymptotic behavior to a logistic chemotaxis model with density-signal governed sensitivity and signal absorption, preprint, arXiv: 1806.09914v1.

[8]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[9]

K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81.

[10]

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.

[11]

F. Heihoff, Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source, Z. Angew. Math. Phys., 71 (2020), Article number: 80. doi: 10.1007/s00033-020-01304-w.

[12]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[13]

T. HillenK. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.  doi: 10.1142/S0218202512500480.

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D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 

[15]

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.

[16] M. Isenbach, Chemotaxis, Imperial College Press, London, 2004. 
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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.

[18]

Z. Jia and Z. Yang, Global existence to a chemotaxis-consumption model with nonlinear diffusion and singular sensitivity, Applicable Analysis, 98 (2019), 2916-2929.  doi: 10.1080/00036811.2018.1478083.

[19]

E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal., Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012.

[20]

E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0.

[21]

J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.

[22]

J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007.

[23]

J. Lankeit and G. Viglialoro, Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity, Acta Appl. Math., 167 (2020), 75-97.  doi: 10.1007/s10440-019-00269-x.

[24]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: global solvability for large nonradial data, Nonlinear Differ. Equ. Appl., 24 (2017), Article number: 49. doi: 10.1007/s00030-017-0472-8.

[25]

B. Liu and G. Ren, Global existence and asymptotic behavior in a three-dimensional two-species chemotaxis-Stokes system with tensor-valued sensitivity, J. Korean Math. Soc., 57 (2020), 215-247.  doi: 10.4134/JKMS.j190028.

[26]

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.

[27]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 26 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[28]

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.

[29]

G. Ren, Boundedness and stabilization in a two-species chemotaxis system with logistic source, Z. Angew. Math. Phys., 77 (2020), 177.

[30]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal.: Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.

[31]

G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal.: Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.

[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]

G. Ren and B. Liu, Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source, Commun. Pure Appl. Anal., 19 (2020), 3843-3883.  doi: 10.3934/cpaa.2020170.

[34]

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.

[35]

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.

[36]

N. Rodriguez and M. Winkler, Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation, Math. Models Methods Appl. Sci., 30 (2020), 2105-2137.  doi: 10.1142/S0218202520500396.

[37]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal., Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.

[38]

G. Viglialoro, Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Appl. Math. Lett., 91 (2019), 121-127.  doi: 10.1016/j.aml.2018.12.012.

[39]

Q. WangD. Wang and Y. Feng, Global well-posedness and uniform boundedness of urban crime models: One-dimensional case, J. Differential Equations, 269 (2020), 6216-6235.  doi: 10.1016/j.jde.2020.04.035.

[40]

W. Wang, The logistic chemotaxis system with singular sensitivity and signal absorption in dimension two, Nonlinear Anal.: Real World Appl., 50 (2019), 532-561.  doi: 10.1016/j.nonrwa.2019.06.001.

[41]

W. WangY. Li and H. Yu, Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3663-3669.  doi: 10.3934/dcdsb.2017147.

[42]

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.

[43]

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.

[44]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.

[45]

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.

[46]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption global large-data solutions and their relaxation properties, Math. Models Meth. Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

[47]

M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differential Equations, 264 (2018), 2310-2350.  doi: 10.1016/j.jde.2017.10.029.

[48]

M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Functional Analysis, 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009.

[49]

M. Winkler, Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation, Ann. Inst. H. Poincaré–Anal. Non Linéaire, 36 (2019), 1747-1790.  doi: 10.1016/j.anihpc.2019.02.004.

[50]

M. Winkler and T. Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Anal., 170 (2018), 123-141.  doi: 10.1016/j.na.2018.01.002.

[51]

J. Yan and Y. Li, Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Anal., 176 (2018), 288-302.  doi: 10.1016/j.na.2018.06.016.

[52]

X. Zhao, Boundedness to a logistic chemotaxis system with singular sensitivity, preprint, arXiv: 2003.03016.

[53]

X. Zhao and S. Zheng, Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 443 (2016), 445-452.  doi: 10.1016/j.jmaa.2016.05.036.

[54]

X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), 1-13.  doi: 10.1007/s00033-016-0749-5.

[55]

X. Zhao and S. Zheng, Global existence and asymptotic behavior to a chemotaxis-consumption system with singular sensitivity and logistic source, Nonlinear Anal., Real World Appl., 42 (2018), 120-139.  doi: 10.1016/j.nonrwa.2017.12.007.

[56]

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

[57]

J. Zheng, Boundedness and large time behavior in a higher-dimensional Keller-Segel system with singular sensitivity and logistic source, preprint, arXiv: 1812.02355v4.

[58]

P. ZhengC. MuR. Willie and X. Hu, Global asymptotic stability of steady states in a chemotaxis-growth system with singular sensitivity, Comput. Math. Appl., 75 (2018), 1667-1675.  doi: 10.1016/j.camwa.2017.11.032.

[59]

A. Zhigun, Generalised supersolutions with mass control for the Keller-Segel system with logarithmic sensitivity, J. Math. Anal. Appl., 467 (2018), 1270-1286.  doi: 10.1016/j.jmaa.2018.08.001.

show all references

References:
[1]

J. AhnK. Kang and J. Lee, Eventual smoothness and stabilization of global weak solutions in parabolic-elliptic chemotaxis systems with logarithmic sensitivity, Nonlinear Anal: Real World Appl., 49 (2019), 312-330.  doi: 10.1016/j.nonrwa.2019.03.012.

[2]

H. Amann, Dynamic theory of quasilinear parabolic systems III: Global existence, Math. Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Meth. Appl. Sci., 25 (2015), 1663-1763.  doi: 10.1142/S021820251550044X.

[4]

T. Black, Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 119-137.  doi: 10.3934/dcdss.2020007.

[5]

L. Chen, F. Kong and Q. Wang, Global and exponential attractor of the repulsive Keller-Segel model with logarithmic sensitivity, Euro. J. Appl. Math., (2020), 1–19. doi: 10.1017/S0956792520000194.

[6]

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

[7]

M. Ding and X. Zhao, Global existence, boundedness and asymptotic behavior to a logistic chemotaxis model with density-signal governed sensitivity and signal absorption, preprint, arXiv: 1806.09914v1.

[8]

K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684.  doi: 10.1016/j.jmaa.2014.11.045.

[9]

K. Fujie and T. Senba, Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 81-102.  doi: 10.3934/dcdsb.2016.21.81.

[10]

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.

[11]

F. Heihoff, Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source, Z. Angew. Math. Phys., 71 (2020), Article number: 80. doi: 10.1007/s00033-020-01304-w.

[12]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[13]

T. HillenK. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Math. Models Methods Appl. Sci., 23 (2013), 165-198.  doi: 10.1142/S0218202512500480.

[14]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165. 

[15]

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.

[16] M. Isenbach, Chemotaxis, Imperial College Press, London, 2004. 
[17]

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.

[18]

Z. Jia and Z. Yang, Global existence to a chemotaxis-consumption model with nonlinear diffusion and singular sensitivity, Applicable Analysis, 98 (2019), 2916-2929.  doi: 10.1080/00036811.2018.1478083.

[19]

E. Lankeit and J. Lankeit, Classical solutions to a logistic chemotaxis model with singular sensitivity and signal absorption, Nonlinear Anal., Real World Appl., 46 (2019), 421-445.  doi: 10.1016/j.nonrwa.2018.09.012.

[20]

E. Lankeit and J. Lankeit, On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms, Nonlinearity, 32 (2019), 1569-1596.  doi: 10.1088/1361-6544/aaf8c0.

[21]

J. Lankeit, A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 39 (2016), 394-404.  doi: 10.1002/mma.3489.

[22]

J. Lankeit, Locally bounded global solutions to a chemotaxis consumption model with singular sensitivity and nonlinear diffusion, J. Differential Equations, 262 (2017), 4052-4084.  doi: 10.1016/j.jde.2016.12.007.

[23]

J. Lankeit and G. Viglialoro, Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity, Acta Appl. Math., 167 (2020), 75-97.  doi: 10.1007/s10440-019-00269-x.

[24]

J. Lankeit and M. Winkler, A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: global solvability for large nonradial data, Nonlinear Differ. Equ. Appl., 24 (2017), Article number: 49. doi: 10.1007/s00030-017-0472-8.

[25]

B. Liu and G. Ren, Global existence and asymptotic behavior in a three-dimensional two-species chemotaxis-Stokes system with tensor-valued sensitivity, J. Korean Math. Soc., 57 (2020), 215-247.  doi: 10.4134/JKMS.j190028.

[26]

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.

[27]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theoret. Biol., 26 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.

[28]

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.

[29]

G. Ren, Boundedness and stabilization in a two-species chemotaxis system with logistic source, Z. Angew. Math. Phys., 77 (2020), 177.

[30]

G. Ren and B. Liu, Global boundedness and asymptotic behavior in a two-species chemotaxis-competition system with two signals, Nonlinear Anal.: Real World Appl., 48 (2019), 288-325.  doi: 10.1016/j.nonrwa.2019.01.017.

[31]

G. Ren and B. Liu, Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source, Nonlinear Anal.: Real World Appl., 46 (2019), 545-582.  doi: 10.1016/j.nonrwa.2018.09.020.

[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]

G. Ren and B. Liu, Global boundedness of solutions to a chemotaxis-fluid system with singular sensitivity and logistic source, Commun. Pure Appl. Anal., 19 (2020), 3843-3883.  doi: 10.3934/cpaa.2020170.

[34]

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.

[35]

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.

[36]

N. Rodriguez and M. Winkler, Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation, Math. Models Methods Appl. Sci., 30 (2020), 2105-2137.  doi: 10.1142/S0218202520500396.

[37]

C. Stinner and M. Winkler, Global weak solutions in a chemotaxis system with large singular sensitivity, Nonlinear Anal., Real World Appl., 12 (2011), 3727-3740.  doi: 10.1016/j.nonrwa.2011.07.006.

[38]

G. Viglialoro, Global existence in a two-dimensional chemotaxis-consumption model with weakly singular sensitivity, Appl. Math. Lett., 91 (2019), 121-127.  doi: 10.1016/j.aml.2018.12.012.

[39]

Q. WangD. Wang and Y. Feng, Global well-posedness and uniform boundedness of urban crime models: One-dimensional case, J. Differential Equations, 269 (2020), 6216-6235.  doi: 10.1016/j.jde.2020.04.035.

[40]

W. Wang, The logistic chemotaxis system with singular sensitivity and signal absorption in dimension two, Nonlinear Anal.: Real World Appl., 50 (2019), 532-561.  doi: 10.1016/j.nonrwa.2019.06.001.

[41]

W. WangY. Li and H. Yu, Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3663-3669.  doi: 10.3934/dcdsb.2017147.

[42]

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.

[43]

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.

[44]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Math. Methods Appl. Sci., 34 (2011), 176-190.  doi: 10.1002/mma.1346.

[45]

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.

[46]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption global large-data solutions and their relaxation properties, Math. Models Meth. Appl. Sci., 26 (2016), 987-1024.  doi: 10.1142/S0218202516500238.

[47]

M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differential Equations, 264 (2018), 2310-2350.  doi: 10.1016/j.jde.2017.10.029.

[48]

M. Winkler, A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization, J. Functional Analysis, 276 (2019), 1339-1401.  doi: 10.1016/j.jfa.2018.12.009.

[49]

M. Winkler, Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation, Ann. Inst. H. Poincaré–Anal. Non Linéaire, 36 (2019), 1747-1790.  doi: 10.1016/j.anihpc.2019.02.004.

[50]

M. Winkler and T. Yokota, Stabilization in the logarithmic Keller-Segel system, Nonlinear Anal., 170 (2018), 123-141.  doi: 10.1016/j.na.2018.01.002.

[51]

J. Yan and Y. Li, Global generalized solutions to a Keller-Segel system with nonlinear diffusion and singular sensitivity, Nonlinear Anal., 176 (2018), 288-302.  doi: 10.1016/j.na.2018.06.016.

[52]

X. Zhao, Boundedness to a logistic chemotaxis system with singular sensitivity, preprint, arXiv: 2003.03016.

[53]

X. Zhao and S. Zheng, Global boundedness of solutions in a parabolic-parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 443 (2016), 445-452.  doi: 10.1016/j.jmaa.2016.05.036.

[54]

X. Zhao and S. Zheng, Global boundedness to a chemotaxis system with singular sensitivity and logistic source, Z. Angew. Math. Phys., 68 (2017), 1-13.  doi: 10.1007/s00033-016-0749-5.

[55]

X. Zhao and S. Zheng, Global existence and asymptotic behavior to a chemotaxis-consumption system with singular sensitivity and logistic source, Nonlinear Anal., Real World Appl., 42 (2018), 120-139.  doi: 10.1016/j.nonrwa.2017.12.007.

[56]

X. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis 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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