Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source

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May 2015, 35(5): 2299-2323. doi: 10.3934/dcds.2015.35.2299

Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source

Pan Zheng ^1, , Chunlai Mu ^2, and Xuegang Hu ^1,

1.	College of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065, China, China
2.	College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received July 2014 Revised August 2014 Published December 2014

Abstract
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This paper deals with a parabolic-elliptic chemotaxis system with generalized volume-filling effect and logistic source \begin{eqnarray*} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(\psi(u)\nabla v)+f(u), &(x,t)\in \Omega\times (0,\infty), \\ &0=\Delta v-m(t)+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi>0$, $m(t)=\frac{1}{|\Omega|}\int_{\Omega}u(x,t)dx$, the nonlinear diffusivity $\varphi(u)$ and chemosensitivity $\psi(u)$ are supposed to extend the prototypes $$\varphi(u)=(u+1)^{-p},\quad \psi(u)=u(u+1)^{q-1}$$ with $p\geq0$, $q\in \mathbb{R}$, and $f(u)$ is assumed to generalize the standard logistic function $$f(u)=\lambda u-\mu u^{k},\;\text{with}\;\;\lambda\geq 0,\mu>0\;\text{and}\;k>1.$$ Under some different suitable assumptions on the nonlinearities $\varphi(u), \psi(u)$ and logistic source $f(u)$, we study the global boundedness and finite-time blow-up of solutions for the problem.

Keywords: finite-time blow-up, logistic source., chemotaxis, Boundedness.

Mathematics Subject Classification: Primary: 35K55, 35B35; Secondary: 35B4.

Citation: Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299

References:

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logistic source, C. R. Acad. Sci. Paris, Ser. I, 351 (2013), 585-591. doi: 10.1016/j.crma.2013.07.027.
[3]	J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038.
[4]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[5]	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.
[6]	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.
[7]	T. Cieślak and P. Laurençot, Finite-time blow-up for one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H.Poincaré Anal. Non Linéarire, 27 (2010), 437-446. doi: 10.1016/j.anihpc.2009.11.016.
[8]	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.
[9]	T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146. doi: 10.1007/s10440-013-9832-5.
[10]	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.
[11]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[12]	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.
[13]	D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478. doi: 10.1007/s002850100134.
[14]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.
[15]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165.
[16]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69.
[17]	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.
[18]	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.
[19]	S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domain, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.
[20]	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.
[21]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[22]	J. Lankeit, Chemotaxis can prevent thresholds on population density, arXiv:1403.1837v1.
[23]	C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642. doi: 10.1016/j.nonrwa.2012.10.022.
[24]	E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044.
[25]	L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., (3) 20 (1966), 733-737.
[26]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[27]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
[28]	C. S. Patlak, Random walk with persistence and external bias, Bull. Math. BiophyS., 15 (1953), 311-338. doi: 10.1007/BF02476407.
[29]	Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.
[30]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[31]	Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543. doi: 10.1016/j.jde.2011.07.010.
[32]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[33]	R. Temam, Infinite-dimensional Dynamical Dystems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[34]	L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789.
[35]	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.
[36]	Z. A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom., 5 (2010), 173-190. doi: 10.1051/mmnp/20105311.
[37]	Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp. doi: 10.1063/1.2766864.
[38]	Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001.
[39]	Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.
[40]	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.
[41]	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.
[42]	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.
[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, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838.
[45]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.
[46]	M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071.
[47]	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.
[48]	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.
[49]	D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal., 73 (2010), 338-349. doi: 10.1016/j.na.2010.02.047.
[50]	D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106.
[51]	P. Zheng, C. Mu and L. Wang, Finite-time blow-up and global boundedness for a quasilinear parabolic-elliptic chemotaxis system with logistic source, preprint.

show all references

References:

[1]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.
[2]	K. Baghaei and M. Hesaaraki, Global existence and boundedness of classical solutions for a chemotaxis model with logistic source, C. R. Acad. Sci. Paris, Ser. I, 351 (2013), 585-591. doi: 10.1016/j.crma.2013.07.027.
[3]	J. Burczak, T. Cieślak and C. Morales-Rodrigo, Global existence vs. blow-up in a fully parabolic quasilinear 1D Keller-Segel system, Nonlinear Anal., 75 (2012), 5215-5228. doi: 10.1016/j.na.2012.04.038.
[4]	V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl., 86 (2006), 155-175. doi: 10.1016/j.matpur.2006.04.002.
[5]	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.
[6]	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.
[7]	T. Cieślak and P. Laurençot, Finite-time blow-up for one-dimensional quasilinear parabolic-parabolic chemotaxis system, Ann. Inst. H.Poincaré Anal. Non Linéarire, 27 (2010), 437-446. doi: 10.1016/j.anihpc.2009.11.016.
[8]	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.
[9]	T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2, Acta Appl. Math., 129 (2014), 135-146. doi: 10.1007/s10440-013-9832-5.
[10]	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.
[11]	A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[12]	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.
[13]	D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model, J. Math. Biol., 44 (2002), 463-478. doi: 10.1007/s002850100134.
[14]	D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.
[15]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103-165.
[16]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51-69.
[17]	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.
[18]	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.
[19]	S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolic-parabolic type on non-convex bounded domain, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028.
[20]	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.
[21]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.
[22]	J. Lankeit, Chemotaxis can prevent thresholds on population density, arXiv:1403.1837v1.
[23]	C. Mu, L. Wang, P. Zheng and Q. Zhang, Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal. Real World Appl., 14 (2013), 1634-1642. doi: 10.1016/j.nonrwa.2012.10.022.
[24]	E. Nakaguchi and K. Osaki, Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation, Nonlinear Anal., 74 (2011), 286-297. doi: 10.1016/j.na.2010.08.044.
[25]	L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., (3) 20 (1966), 733-737.
[26]	K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144. doi: 10.1016/S0362-546X(01)00815-X.
[27]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
[28]	C. S. Patlak, Random walk with persistence and external bias, Bull. Math. BiophyS., 15 (1953), 311-338. doi: 10.1007/BF02476407.
[29]	Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36. doi: 10.1142/S0218202512500443.
[30]	Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715. doi: 10.1016/j.jde.2011.08.019.
[31]	Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543. doi: 10.1016/j.jde.2011.07.010.
[32]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[33]	R. Temam, Infinite-dimensional Dynamical Dystems in Mechanics and Physics, 2nd ed., Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[34]	L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 34 (2014), 789-802. doi: 10.3934/dcds.2014.34.789.
[35]	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.
[36]	Z. A. Wang, On chemotaxis models with cell population interactions, Math. Model. Nat. Phenom., 5 (2010), 173-190. doi: 10.1051/mmnp/20105311.
[37]	Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model, Chaos, 17 (2007), 037108, 13pp. doi: 10.1063/1.2766864.
[38]	Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001.
[39]	Z. A. Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972.
[40]	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.
[41]	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.
[42]	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.
[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, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283 (2010), 1664-1673. doi: 10.1002/mana.200810838.
[45]	M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.
[46]	M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties, J. Math. Anal. Appl., 348 (2008), 708-729. doi: 10.1016/j.jmaa.2008.07.071.
[47]	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.
[48]	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.
[49]	D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonlinear Anal., 73 (2010), 338-349. doi: 10.1016/j.na.2010.02.047.
[50]	D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106.
[51]	P. Zheng, C. Mu and L. Wang, Finite-time blow-up and global boundedness for a quasilinear parabolic-elliptic chemotaxis system with logistic source, preprint.

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