2015, 35(8): 3503-3531. doi: 10.3934/dcds.2015.35.3503

Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source

1. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731

Received  November 2014 Revised  December 2014 Published  February 2015

In this paper, we investigate the quasilinear Keller-Segel equations (q-K-S): \[ \left\{ \begin{split} &n_t=\nabla\cdot\big(D(n)\nabla n\big)-\nabla\cdot\big(\chi(n)\nabla c\big)+\mathcal{R}(n), \qquad x\in\Omega,\,t>0,\\ &\varrho c_t=\Delta c-c+n, \qquad x\in\Omega,\,t>0, \end{split} \right. \] under homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^N$. For both $\varrho=0$ (parabolic-elliptic case) and $\varrho>0$ (parabolic-parabolic case), we will show the global-in-time existence and uniform-in-time boundedness of solutions to equations (q-K-S) with both non-degenerate and degenerate diffusions on the non-convex domain $\Omega$, which provide a supplement to the dichotomy boundedness vs. blow-up in parabolic-elliptic/parabolic-parabolic chemotaxis equations with degenerate diffusion, nonlinear sensitivity and logistic source. In particular, we improve the recent results obtained by Wang-Li-Mu (2014, Disc. Cont. Dyn. Syst.) and Wang-Mu-Zheng (2014, J. Differential Equations).
Citation: Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503
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.

[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.

[3]

J. Burczak, T. Cieslak 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.

[4]

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.

[5]

T. Cieslak, Quasilinear nonuniformly parabolic system modelling chemotaxis,, J. Math. Anal. Appl., 326 (2007), 1410. doi: 10.1016/j.jmaa.2006.03.080.

[6]

T. Cieslak and P. Laurencot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. I. H. Poincaré AN, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016.

[7]

T. Cieslak and C. Stinner, Finite-time blow up 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.

[8]

T. Cieslak 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.

[9]

T. Cieslak 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.

[10]

A. Friedman, Partial Differential Equations,, Holt, (1969).

[11]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, in Harmonic Analysis and Nonlinear Partial Differential Equations, 26 (2011), 159.

[12]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations,, European Mathematical Society, (2008).

[13]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.

[14]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Adv. Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721.

[15]

T. Hillen and K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501.

[16]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3.

[17]

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.

[18]

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.

[19]

S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Math. Methods Appl. Sci., 36 (2013), 745. doi: 10.1002/mma.2622.

[20]

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. doi: 10.1016/j.jde.2014.01.028.

[21]

W. Jager and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819. doi: 10.1090/S0002-9947-1992-1046835-6.

[22]

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.

[23]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system,, Ann. I. H. Poincaré AN, 31 (2014), 851. doi: 10.1016/j.anihpc.2013.07.007.

[24]

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.

[25]

T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.

[26]

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.

[27]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theoret. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5.

[28]

L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733.

[29]

T. Senba, Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains,, Funkcialaj Ekvacioj, 48 (2005), 247. doi: 10.1619/fesi.48.247.

[30]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time,, Methods Appl. Anal., 8 (2001), 349.

[31]

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.

[32]

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.

[33]

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.

[34]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003.

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, 68 (1997).

[36]

L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source,, Discrete and Continuous Dynamical Systems, 34 (2014), 789. doi: 10.3934/dcds.2014.34.789.

[37]

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.

[38]

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.

[39]

Z. A. Wang, On chemotaxis models with cell population interactions,, Math. Model. Nat. Phenom., 5 (2010), 173. doi: 10.1051/mmnp/20105311.

[40]

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.

[41]

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.

[42]

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.

[43]

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.

[44]

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.

[45]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146.

[46]

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.

[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. doi: 10.1016/j.na.2009.07.045.

[48]

D. Wrzosek, Global attractor for a chemotaxis model with prevention of over-crowding,, Nonlinear Anal., 59 (2004), 1293. doi: 10.1016/j.na.2004.08.015.

show all references

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.

[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.

[3]

J. Burczak, T. Cieslak 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.

[4]

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.

[5]

T. Cieslak, Quasilinear nonuniformly parabolic system modelling chemotaxis,, J. Math. Anal. Appl., 326 (2007), 1410. doi: 10.1016/j.jmaa.2006.03.080.

[6]

T. Cieslak and P. Laurencot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. I. H. Poincaré AN, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016.

[7]

T. Cieslak and C. Stinner, Finite-time blow up 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.

[8]

T. Cieslak 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.

[9]

T. Cieslak 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.

[10]

A. Friedman, Partial Differential Equations,, Holt, (1969).

[11]

H. Hajaiej, L. Molinet, T. Ozawa and B. Wang, Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized boson equations,, in Harmonic Analysis and Nonlinear Partial Differential Equations, 26 (2011), 159.

[12]

D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations,, European Mathematical Society, (2008).

[13]

M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.

[14]

T. Hillen and K. J. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding,, Adv. Appl. Math., 26 (2001), 280. doi: 10.1006/aama.2001.0721.

[15]

T. Hillen and K. Painter, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Q., 10 (2002), 501.

[16]

T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3.

[17]

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.

[18]

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.

[19]

S. Ishida, T. Ono and T. Yokota, Possibility of the existence of blow-up solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Math. Methods Appl. Sci., 36 (2013), 745. doi: 10.1002/mma.2622.

[20]

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. doi: 10.1016/j.jde.2014.01.028.

[21]

W. Jager and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819. doi: 10.1090/S0002-9947-1992-1046835-6.

[22]

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.

[23]

N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system,, Ann. I. H. Poincaré AN, 31 (2014), 851. doi: 10.1016/j.anihpc.2013.07.007.

[24]

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.

[25]

T. Nagai, Blow-up of radially symmetric solutions of a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.

[26]

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.

[27]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, J. Theoret. Biol., 42 (1973), 63. doi: 10.1016/0022-5193(73)90149-5.

[28]

L. Nirenberg, An extended interpolation inequality,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (1966), 733.

[29]

T. Senba, Blowup behavior of radial solutions to Jager-Luckhaus system in high dimensional domains,, Funkcialaj Ekvacioj, 48 (2005), 247. doi: 10.1619/fesi.48.247.

[30]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time,, Methods Appl. Anal., 8 (2001), 349.

[31]

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.

[32]

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.

[33]

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.

[34]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source,, Comm. Partial Differential Equations, 32 (2007), 849. doi: 10.1080/03605300701319003.

[35]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, 68 (1997).

[36]

L. Wang, Y. Li and C. Mu, Boundedness in a parabolic-parabolic quasilinear chemotaxis system with logistic source,, Discrete and Continuous Dynamical Systems, 34 (2014), 789. doi: 10.3934/dcds.2014.34.789.

[37]

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.

[38]

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.

[39]

Z. A. Wang, On chemotaxis models with cell population interactions,, Math. Model. Nat. Phenom., 5 (2010), 173. doi: 10.1051/mmnp/20105311.

[40]

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.

[41]

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.

[42]

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.

[43]

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.

[44]

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.

[45]

M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12. doi: 10.1002/mma.1146.

[46]

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.

[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. doi: 10.1016/j.na.2009.07.045.

[48]

D. Wrzosek, Global attractor for a chemotaxis model with prevention of over-crowding,, Nonlinear Anal., 59 (2004), 1293. doi: 10.1016/j.na.2004.08.015.

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