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Dynamic observers for unknown populations
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 |
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.
References:
[1] |
J. Ahn, K. 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. Bellomo, A. Bellouquid, Y. 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. Ding, W. 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. 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. |
[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. 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. |
[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. Hillen, K. 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. 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. |
[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. |
[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. 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. |
[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. Wang, D. 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. Wang, Y. 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. Google Scholar |
[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. Google Scholar |
[58] |
P. Zheng, C. Mu, R. 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. Ahn, K. 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. Bellomo, A. Bellouquid, Y. 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. Ding, W. 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. 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. |
[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. 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. |
[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. Hillen, K. 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. 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. |
[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. |
[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. 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. |
[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. Wang, D. 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. Wang, Y. 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. Google Scholar |
[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. Google Scholar |
[58] |
P. Zheng, C. Mu, R. 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. |
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