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A quasilinear parabolic-parabolic chemotaxis model with logistic source and singular sensitivity
College of Mathematics and Information, China West Normal University, NanChong 637000, China |
$ \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*} $ |
$ \Omega\subset\mathbb{R}^{n} $ |
$ n\geq2 $ |
$ \chi>0 $ |
$ \mu>0 $ |
$ D(u) $ |
$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $ |
$ m_{*} $ |
$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $ |
$ t\geq0 $ |
$ (1, 1) $ |
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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