# American Institute of Mathematical Sciences

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

 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

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.
Citation: Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
##### References:
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Appl., 412 (2014), 181. doi: 10.1016/j.jmaa.2013.10.061. Google Scholar [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. doi: 10.1002/mma.2992. Google Scholar [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. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar [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. doi: 10.1016/j.jde.2012.01.045. Google Scholar [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. doi: 10.1007/s10440-013-9832-5. 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Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103. Google Scholar [16] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51. Google Scholar [17] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [18] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar [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. doi: 10.1016/j.jde.2014.01.028. Google Scholar [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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [21] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [22] J. Lankeit, Chemotaxis can prevent thresholds on population density,, , (). Google Scholar [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. doi: 10.1016/j.nonrwa.2012.10.022. Google Scholar [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. doi: 10.1016/j.na.2010.08.044. Google Scholar [25] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733. Google Scholar [26] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar [27] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar [28] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. BiophyS., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar [29] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [30] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019. Google Scholar [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. doi: 10.1016/j.jde.2011.07.010. Google Scholar [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [33] R. Temam, Infinite-dimensional Dynamical Dystems in Mechanics and Physics,, 2nd ed., (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [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. doi: 10.3934/dcds.2014.34.789. Google Scholar [35] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source,, J. Differential Equations, 256 (2014), 1847. doi: 10.1016/j.jde.2013.12.007. Google Scholar [36] Z. A. Wang, On chemotaxis models with cell population interactions,, Math. Model. Nat. Phenom., 5 (2010), 173. doi: 10.1051/mmnp/20105311. Google Scholar [37] Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007). doi: 10.1063/1.2766864. Google Scholar [38] Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect,, Nonlinearity, 24 (2011), 3279. doi: 10.1088/0951-7715/24/12/001. Google Scholar [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. doi: 10.1137/110853972. Google Scholar [40] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction,, J. Math. Anal. Appl., 384 (2011), 261. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar [41] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar [42] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [43] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [44] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664. doi: 10.1002/mana.200810838. Google Scholar [45] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146. Google Scholar [46] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [47] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?,, J. Nonlinear Sci., 24 (2014), 809. doi: 10.1007/s00332-014-9205-x. Google Scholar [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. doi: 10.1016/j.na.2009.07.045. Google Scholar [49] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion,, Nonlinear Anal., 73 (2010), 338. doi: 10.1016/j.na.2010.02.047. Google Scholar [50] D. Wrzosek, Volume filling effect in modelling chemotaxis,, Math. Model. Nat. Phenom., 5 (2010), 123. doi: 10.1051/mmnp/20105106. Google Scholar [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., (). Google Scholar

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##### References:
 [1] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations,, Comm. Partial Differential Equations, 4 (1979), 827. doi: 10.1080/03605307908820113. Google Scholar [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, 351 (2013), 585. doi: 10.1016/j.crma.2013.07.027. Google Scholar [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. doi: 10.1016/j.na.2012.04.038. Google Scholar [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. doi: 10.1016/j.matpur.2006.04.002. Google Scholar [5] X. Cao, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with logistic source,, J. Math. Anal. Appl., 412 (2014), 181. doi: 10.1016/j.jmaa.2013.10.061. Google Scholar [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. doi: 10.1002/mma.2992. Google Scholar [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. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar [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. doi: 10.1016/j.jde.2012.01.045. Google Scholar [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. doi: 10.1007/s10440-013-9832-5. Google Scholar [10] T. Cieślak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis,, Nonlinearity, 21 (2008), 1057. doi: 10.1088/0951-7715/21/5/009. Google Scholar [11] A. Friedman, Partial Differential Equations,, Holt, (1969). Google Scholar [12] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar [13] D. Horstmann, On the existence of radially symmetric blow-up solutions for the Keller-Segel model,, J. Math. Biol., 44 (2002), 463. doi: 10.1007/s002850100134. Google Scholar [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. doi: 10.1007/s00332-010-9082-x. Google Scholar [15] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jahresber. Deutsch. Math. -Verein., 105 (2003), 103. Google Scholar [16] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences II,, Jahresber. Deutsch. Math. -Verein., 106 (2004), 51. Google Scholar [17] D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159. doi: 10.1017/S0956792501004363. Google Scholar [18] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar [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. doi: 10.1016/j.jde.2014.01.028. Google Scholar [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. doi: 10.1090/S0002-9947-1992-1046835-6. Google Scholar [21] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar [22] J. Lankeit, Chemotaxis can prevent thresholds on population density,, , (). Google Scholar [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. doi: 10.1016/j.nonrwa.2012.10.022. Google Scholar [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. doi: 10.1016/j.na.2010.08.044. Google Scholar [25] L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733. Google Scholar [26] K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Exponential attractor for a chemotaxis-growth system of equations,, Nonlinear Anal., 51 (2002), 119. doi: 10.1016/S0362-546X(01)00815-X. Google Scholar [27] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501. Google Scholar [28] C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. BiophyS., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar [29] Y. Tao and Z. A. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models Methods Appl. Sci., 23 (2013), 1. doi: 10.1142/S0218202512500443. Google Scholar [30] Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692. doi: 10.1016/j.jde.2011.08.019. Google Scholar [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. doi: 10.1016/j.jde.2011.07.010. Google Scholar [32] J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003. Google Scholar [33] R. Temam, Infinite-dimensional Dynamical Dystems in Mechanics and Physics,, 2nd ed., (1997). doi: 10.1007/978-1-4612-0645-3. Google Scholar [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. doi: 10.3934/dcds.2014.34.789. Google Scholar [35] L. Wang, C. Mu and P. Zheng, On a quasilinear parabolic-elliptic chemotaxis system with logistic source,, J. Differential Equations, 256 (2014), 1847. doi: 10.1016/j.jde.2013.12.007. Google Scholar [36] Z. A. Wang, On chemotaxis models with cell population interactions,, Math. Model. Nat. Phenom., 5 (2010), 173. doi: 10.1051/mmnp/20105311. Google Scholar [37] Z. A. Wang and T. Hillen, Classical solutions and pattern formation for a volume filling chemotaxis model,, Chaos, 17 (2007). doi: 10.1063/1.2766864. Google Scholar [38] Z. A. Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect,, Nonlinearity, 24 (2011), 3279. doi: 10.1088/0951-7715/24/12/001. Google Scholar [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. doi: 10.1137/110853972. Google Scholar [40] M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction,, J. Math. Anal. Appl., 384 (2011), 261. doi: 10.1016/j.jmaa.2011.05.057. Google Scholar [41] M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748. doi: 10.1016/j.matpur.2013.01.020. Google Scholar [42] M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889. doi: 10.1016/j.jde.2010.02.008. Google Scholar [43] M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source,, Comm. Partial Differential Equations, 35 (2010), 1516. doi: 10.1080/03605300903473426. Google Scholar [44] M. Winkler, Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity,, Math. Nachr., 283 (2010), 1664. doi: 10.1002/mana.200810838. Google Scholar [45] M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146. Google Scholar [46] M. Winkler, Chemotaxis with logistic source: Very weak global solutions and their boundedness properties,, J. Math. Anal. Appl., 348 (2008), 708. doi: 10.1016/j.jmaa.2008.07.071. Google Scholar [47] M. Winkler, How far can chemotactic cross-diffusion enforce exceeding carrying capacities?,, J. Nonlinear Sci., 24 (2014), 809. doi: 10.1007/s00332-014-9205-x. Google Scholar [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. doi: 10.1016/j.na.2009.07.045. Google Scholar [49] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion,, Nonlinear Anal., 73 (2010), 338. doi: 10.1016/j.na.2010.02.047. Google Scholar [50] D. Wrzosek, Volume filling effect in modelling chemotaxis,, Math. Model. Nat. Phenom., 5 (2010), 123. doi: 10.1051/mmnp/20105106. Google Scholar [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., (). Google Scholar
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