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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  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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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., Real World Appl., 49 (2019), 286-311.  doi: 10.1016/j.nonrwa.2019.03.009.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  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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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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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  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

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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.  Google Scholar

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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

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

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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

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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

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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.  Google Scholar

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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.  Google Scholar

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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

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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.  Google Scholar

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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

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