# 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, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
##### References:
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Math., 129 (2014), 135-146. 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), 1057-1076. doi: 10.1088/0951-7715/21/5/009.  Google Scholar [11] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar [12] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. 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), 463-478. 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), 231-270. 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), 103-165.  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), 51-69.  Google Scholar [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.  Google Scholar [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.  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), 2993-3010. 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), 819-824. 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), 399-415. 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), 1634-1642. 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), 286-297. doi: 10.1016/j.na.2010.08.044.  Google Scholar [25] L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., (3) 20 (1966), 733-737.  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), 119-144. 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), 501-543.  Google Scholar [28] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. BiophyS., 15 (1953), 311-338. 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), 1-36. 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), 692-715. 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), 2520-2543. 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), 849-877. doi: 10.1080/03605300701319003.  Google Scholar [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.  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), 789-802. 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), 1847-1872. 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), 173-190. 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), 037108, 13pp. 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), 3279-3297. 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), 3502-3525. 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), 261-272. 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), 748-767. 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), 2889-2905. 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), 1516-1537. 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), 1664-1673. 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), 12-24. 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), 708-729. 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), 809-855. 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), 1044-1064. 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), 338-349. 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), 123-147. 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

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.  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, Ser. I, 351 (2013), 585-591. 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), 5215-5228. 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), 155-175. 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), 181-188. 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), 2326-2330. 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), 437-446. 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), 5832-5851. 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), 135-146. 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), 1057-1076. doi: 10.1088/0951-7715/21/5/009.  Google Scholar [11] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.  Google Scholar [12] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. 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), 463-478. 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), 231-270. 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), 103-165.  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), 51-69.  Google Scholar [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.  Google Scholar [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.  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), 2993-3010. 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), 819-824. 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), 399-415. 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), 1634-1642. 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), 286-297. doi: 10.1016/j.na.2010.08.044.  Google Scholar [25] L. Nirenberg, An extended interpolation inequality, Ann. Sc. Norm. Super. Pisa Cl. Sci., (3) 20 (1966), 733-737.  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), 119-144. 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), 501-543.  Google Scholar [28] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. BiophyS., 15 (1953), 311-338. 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), 1-36. 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), 692-715. 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), 2520-2543. 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), 849-877. doi: 10.1080/03605300701319003.  Google Scholar [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.  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), 789-802. 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), 1847-1872. 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), 173-190. 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), 037108, 13pp. 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), 3279-3297. 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), 3502-3525. 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), 261-272. 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), 748-767. 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), 2889-2905. 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. 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