August  2016, 21(6): 1953-1973. doi: 10.3934/dcdsb.2016031

Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system

1. 

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

Received  December 2014 Revised  March 2016 Published  June 2016

In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type \begin{equation*} \left\{ \begin{split} &u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\ &v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\ &w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0 \end{split} \right. \end{equation*} under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).
Citation: Yilong Wang, Zhaoyin Xiang. Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1953-1973. doi: 10.3934/dcdsb.2016031
References:
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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

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T. Hillen and K. Painter, A users guide to PDE models for chemotaxis,, J. Math.Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103. Google Scholar

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D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Deutsch.Math.-Verien, 106 (2004), 51. Google Scholar

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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

[11]

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. Google Scholar

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H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system,, J. Math. Anal. Appl., 422 (2015), 1463. doi: 10.1016/j.jmaa.2014.09.049. Google Scholar

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H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model,, Math. Meth. Appl. Sci., 38 (2015), 444. doi: 10.1002/mma.3080. Google Scholar

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H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis,, J. Differential Equations, 260 (2016), 162. doi: 10.1016/j.jde.2015.08.040. Google Scholar

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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

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X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source,, Discrete Continuous Dynam. Systems, 35 (2015), 3503. doi: 10.3934/dcds.2015.35.3503. Google Scholar

[18]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source,, IMA J. Appl. Math., 81 (2016), 165. doi: 10.1093/imamat/hxv033. Google Scholar

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J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension,, J. Biol. Dynam., 6 (2012), 31. doi: 10.1080/17513758.2011.571722. Google Scholar

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P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system,, Discrete Continuous Dynam. Systems - B, 18 (2013), 2597. doi: 10.3934/dcdsb.2013.18.2597. Google Scholar

[21]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model,, Math. Methods Appl. Sci., 38 (2015), 2537. doi: 10.1002/mma.3240. Google Scholar

[22]

Y. Lou, Y. Tao and M. Winkler, Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal,, SIAM J. Math. Anal., 46 (2014), 1228. doi: 10.1137/130934246. Google Scholar

[23]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimers disease senile plague: Is there a connection?,, Bull. Math. Biol., 65 (2003), 693. doi: 10.1016/S0092-8240(03)00030-2. Google Scholar

[24]

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

[25]

L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa , 20 (1966), 733. Google Scholar

[26]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520. doi: 10.1016/j.jde.2011.07.010. Google Scholar

[31]

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

[32]

L. C. Wang, C. L. 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

[33]

Y. Wang and Z. Xiang, Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system,, Z. Angew. Math. Phys., 66 (2015), 3159. doi: 10.1007/s00033-015-0557-3. Google Scholar

[34]

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

[35]

M. Winkler, A critical exponent in a degenerate parabolic equation,, Math. Methods Appl. Sci., 25 (2002), 911. doi: 10.1002/mma.319. Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

show all references

References:
[1]

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

[2]

T. Cieślak and P. Laurençot, Finite time blow-up for a one-dimensional quasilinear parabolic-parabolic chemotaxis system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 437. doi: 10.1016/j.anihpc.2009.11.016. Google Scholar

[3]

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

[4]

Y. S. Choi and Z. A. Wang, Prevention of blow-up by fast diffusion in chemotaxis,, J. Math. Anal. Appl., 362 (2010), 553. doi: 10.1016/j.jmaa.2009.08.012. Google Scholar

[5]

M. Chuai, W. Zeng, X. Yang, V. Boychenko, J. A. Glazier and C. J. Weijer, Cell movement during chick primitive streak formation,, Dev. Biol., 296 (2006), 137. doi: 10.1016/j.ydbio.2006.04.451. Google Scholar

[6]

M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and repulsive cues direct the formation of the nigro-striatal circuit,, Euro. J. Neurosci., 19 (2004), 831. doi: 10.1111/j.1460-9568.2004.03213.x. Google Scholar

[7]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis,, J. Math.Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103. Google Scholar

[9]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. Deutsch.Math.-Verien, 106 (2004), 51. Google Scholar

[10]

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

[11]

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. Google Scholar

[12]

S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type,, Discrete Continuous Dynam. Systems - B, 18 (2013), 2569. doi: 10.3934/dcdsb.2013.18.2569. Google Scholar

[13]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system,, J. Math. Anal. Appl., 422 (2015), 1463. doi: 10.1016/j.jmaa.2014.09.049. Google Scholar

[14]

H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model,, Math. Meth. Appl. Sci., 38 (2015), 444. doi: 10.1002/mma.3080. Google Scholar

[15]

H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis,, J. Differential Equations, 260 (2016), 162. doi: 10.1016/j.jde.2015.08.040. Google Scholar

[16]

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

[17]

X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source,, Discrete Continuous Dynam. Systems, 35 (2015), 3503. doi: 10.3934/dcds.2015.35.3503. Google Scholar

[18]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source,, IMA J. Appl. Math., 81 (2016), 165. doi: 10.1093/imamat/hxv033. Google Scholar

[19]

J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension,, J. Biol. Dynam., 6 (2012), 31. doi: 10.1080/17513758.2011.571722. Google Scholar

[20]

P. Liu, J. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system,, Discrete Continuous Dynam. Systems - B, 18 (2013), 2597. doi: 10.3934/dcdsb.2013.18.2597. Google Scholar

[21]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model,, Math. Methods Appl. Sci., 38 (2015), 2537. doi: 10.1002/mma.3240. Google Scholar

[22]

Y. Lou, Y. Tao and M. Winkler, Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal,, SIAM J. Math. Anal., 46 (2014), 1228. doi: 10.1137/130934246. Google Scholar

[23]

M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimers disease senile plague: Is there a connection?,, Bull. Math. Biol., 65 (2003), 693. doi: 10.1016/S0092-8240(03)00030-2. Google Scholar

[24]

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

[25]

L. Nirenberg, An extended interpolation inequality,, Ann. Scuola Norm. Sup. Pisa , 20 (1966), 733. Google Scholar

[26]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

Y. Tao and M. Winkler, Eventual smoothness and stabilization of large-data solutions in three-dimensional chemotaxis system with consumption of chemoattractant,, J. Differential Equations, 252 (2012), 2520. doi: 10.1016/j.jde.2011.07.010. Google Scholar

[31]

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

[32]

L. C. Wang, C. L. 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

[33]

Y. Wang and Z. Xiang, Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system,, Z. Angew. Math. Phys., 66 (2015), 3159. doi: 10.1007/s00033-015-0557-3. Google Scholar

[34]

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

[35]

M. Winkler, A critical exponent in a degenerate parabolic equation,, Math. Methods Appl. Sci., 25 (2002), 911. doi: 10.1002/mma.319. Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

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