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A blow-up result for the chemotaxis system with nonlinear signal production and logistic source
College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_t = \Delta u-\chi\nabla\cdot( u\nabla v)+\lambda u-\mu u^{\alpha}, \quad &x\in \Omega, t>0, \\ 0 = \Delta v-\mu (t)+f(u), \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}f(u(\cdot, t)), \quad &x\in \Omega, t>0, \ \end{array} \right. \end{eqnarray*} $ |
$ \Omega = B_R(0)\subset \mathbb{R}^n $ |
$ n\geq 1 $ |
$ R>0 $ |
$ \chi, \lambda, \mu>0 $ |
$ \alpha>1 $ |
$ f $ |
$ f(u)\geq ku^{\kappa} $ |
$ u\geq1, \kappa>0 $ |
$ k>0 $ |
$ \kappa $ |
$ \alpha $ |
$ \begin{equation*} \kappa+1>\alpha\left(\frac{2}{n}+1\right), \end{equation*} $ |
$ (u, v) $ |
$ \kappa = 1 $ |
References:
[1] |
N. Bellomo, A. Belloquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1764.
doi: 10.1142/S021820251550044X. |
[2] |
X. Cao,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[3] |
X. Cao and S. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.
doi: 10.1002/mma.2992. |
[4] |
T. Cieślak 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. |
[5] |
T. Cieślak 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. |
[6] |
M. Fuest, Finite-time blow-up in a two-dimensional Keller-Segel system with an environmental dependent logistic source, Nonlinear Anal. Real World Appl., 52 (2020), 14pp.
doi: 10.1016/j.nonrwa.2019.103022. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[12] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[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] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[15] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[16] |
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. |
[17] |
R. Kowalczyk,
Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[18] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968. |
[19] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[20] |
P. Laurençot and N. Mizoguchi,
Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal Non Linéaire, 34 (2017), 197-220.
doi: 10.1016/j.anihpc.2015.11.002. |
[21] |
X. Li and Z. Xiang,
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.
doi: 10.3934/dcds.2015.35.3503. |
[22] |
Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, J. Math. Anal. Appl., 480 (2019), 18pp.
doi: 10.1016/j.jmaa.2019.123376. |
[23] |
G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148 (1987), 77–99.
doi: 10.1007/BF01774284. |
[24] |
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. |
[25] |
K. Lin, C. Mu and H. Zhong,
A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions, J. Math. Anal. Appl., 464 (2018), 435-455.
doi: 10.1016/j.jmaa.2018.04.015. |
[26] |
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray,
Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719.
doi: 10.1007/BF02461550. |
[27] |
M. R. Myerscough, P. K. Maini and K. J. Painter,
Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26.
doi: 10.1006/bulm.1997.0010. |
[28] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[29] |
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. |
[30] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[31] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[32] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[33] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
|
[34] |
Y. Sugiyama and H. Kunii,
Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[35] |
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. |
[36] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[37] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[38] |
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. |
[39] |
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. |
[40] |
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. |
[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,
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. |
[43] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel
system, J. Math. Pures Appl. (9), 100 (2013), 748–767.
doi: 10.1016/j.matpur.2013.01.020. |
[44] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), 40pp.
doi: 10.1007/s00033-018-0935-8. |
[45] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[46] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[47] |
M. Winkler and K. C. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[48] |
T. Xiang,
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
doi: 10.1016/j.jmaa.2017.11.022. |
[49] |
Q. Zhang and Y. Li,
Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.
doi: 10.1007/s00033-015-0532-z. |
[50] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
[51] |
P. Zheng, C. Mu and X. Hu,
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 2299-2323.
doi: 10.3934/dcds.2015.35.2299. |
show all references
References:
[1] |
N. Bellomo, A. Belloquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1764.
doi: 10.1142/S021820251550044X. |
[2] |
X. Cao,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 412 (2014), 181-188.
doi: 10.1016/j.jmaa.2013.10.061. |
[3] |
X. Cao and S. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, Math. Methods Appl. Sci., 37 (2014), 2326-2330.
doi: 10.1002/mma.2992. |
[4] |
T. Cieślak 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. |
[5] |
T. Cieślak 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. |
[6] |
M. Fuest, Finite-time blow-up in a two-dimensional Keller-Segel system with an environmental dependent logistic source, Nonlinear Anal. Real World Appl., 52 (2020), 14pp.
doi: 10.1016/j.nonrwa.2019.103022. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[12] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[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] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[15] |
K. Kang and A. Stevens,
Blowup and global solutions in a chemotaxis-growth system, Nonlinear Anal., 135 (2016), 57-72.
doi: 10.1016/j.na.2016.01.017. |
[16] |
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. |
[17] |
R. Kowalczyk,
Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[18] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968. |
[19] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[20] |
P. Laurençot and N. Mizoguchi,
Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal Non Linéaire, 34 (2017), 197-220.
doi: 10.1016/j.anihpc.2015.11.002. |
[21] |
X. Li and Z. Xiang,
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 3503-3531.
doi: 10.3934/dcds.2015.35.3503. |
[22] |
Y. Li, Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production, J. Math. Anal. Appl., 480 (2019), 18pp.
doi: 10.1016/j.jmaa.2019.123376. |
[23] |
G. M. Lieberman, Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, Ann. Mat. Pura Appl. (4), 148 (1987), 77–99.
doi: 10.1007/BF01774284. |
[24] |
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. |
[25] |
K. Lin, C. Mu and H. Zhong,
A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions, J. Math. Anal. Appl., 464 (2018), 435-455.
doi: 10.1016/j.jmaa.2018.04.015. |
[26] |
P. K. Maini, M. R. Myerscough, K. H. Winters and J. D. Murray,
Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation, Bull. Math. Biol., 53 (1991), 701-719.
doi: 10.1007/BF02461550. |
[27] |
M. R. Myerscough, P. K. Maini and K. J. Painter,
Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26.
doi: 10.1006/bulm.1997.0010. |
[28] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[29] |
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. |
[30] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[31] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[32] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[33] |
Y. Sugiyama,
Global existence in sub-critical cases and finite time blow-up in super-critical cases to degenerate Keller-Segel systems, Differential Integral Equations, 19 (2006), 841-876.
|
[34] |
Y. Sugiyama and H. Kunii,
Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364.
doi: 10.1016/j.jde.2006.03.003. |
[35] |
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. |
[36] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[37] |
L. Wang, Y. Li and C. Mu,
Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst., 34 (2014), 789-802.
doi: 10.3934/dcds.2014.34.789. |
[38] |
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. |
[39] |
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. |
[40] |
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. |
[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,
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. |
[43] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel
system, J. Math. Pures Appl. (9), 100 (2013), 748–767.
doi: 10.1016/j.matpur.2013.01.020. |
[44] |
M. Winkler, Finite-time blow-up in low-dimensional Keller-Segel systems with logistic-type superlinear degradation, Z. Angew. Math. Phys., 69 (2018), 40pp.
doi: 10.1007/s00033-018-0935-8. |
[45] |
M. Winkler,
A critical blow-up exponent in a chemotaxis system with nonlinear signal production, Nonlinearity, 31 (2018), 2031-2056.
doi: 10.1088/1361-6544/aaaa0e. |
[46] |
M. Winkler,
How far can chemotactic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[47] |
M. Winkler and K. C. Djie,
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064.
doi: 10.1016/j.na.2009.07.045. |
[48] |
T. Xiang,
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
doi: 10.1016/j.jmaa.2017.11.022. |
[49] |
Q. Zhang and Y. Li,
Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.
doi: 10.1007/s00033-015-0532-z. |
[50] |
J. Zheng,
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140.
doi: 10.1016/j.jde.2015.02.003. |
[51] |
P. Zheng, C. Mu and X. Hu,
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source, Discrete Contin. Dyn. Syst., 35 (2015), 2299-2323.
doi: 10.3934/dcds.2015.35.2299. |
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