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 and 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), 5215-5228. doi: 10.1016/j.na.2012.04.038.

[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), 437-446. doi: 10.1016/j.anihpc.2009.11.016.

[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), 5832-5851. doi: 10.1016/j.jde.2012.01.045.

[4]

Y. S. Choi and Z. A. Wang, Prevention of blow-up by fast diffusion in chemotaxis, J. Math. Anal. Appl., 362 (2010), 553-564. doi: 10.1016/j.jmaa.2009.08.012.

[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), 137-149. doi: 10.1016/j.ydbio.2006.04.451.

[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), 831-844. doi: 10.1111/j.1460-9568.2004.03213.x.

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T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math.Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

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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), 103-165.

[9]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch.Math.-Verien, 106 (2004), 51-69.

[10]

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.

[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), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

[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), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569.

[13]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.

[14]

H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080.

[15]

H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040.

[16]

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.

[17]

X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Continuous Dynam. Systems, 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.

[18]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198. doi: 10.1093/imamat/hxv033.

[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), 31-41. doi: 10.1080/17513758.2011.571722.

[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), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.

[21]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546. doi: 10.1002/mma.3240.

[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), 1228-1262. doi: 10.1137/130934246.

[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), 693-730. doi: 10.1016/S0092-8240(03)00030-2.

[24]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

[25]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa , 20 (1966), 733-737.

[26]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.

[27]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.

[28]

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.

[29]

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.

[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), 2520-2543. doi: 10.1016/j.jde.2011.07.010.

[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), 789-802. doi: 10.3934/dcds.2014.34.789.

[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), 1847-1872. doi: 10.1016/j.jde.2013.12.007.

[33]

Y. Wang and Z. Xiang, Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 3159-3179. doi: 10.1007/s00033-015-0557-3.

[34]

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.

[35]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319.

[36]

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.

[37]

M. Winkler, Does a volume-filling effect always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.

[38]

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.

[39]

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.

[40]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

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), 5215-5228. doi: 10.1016/j.na.2012.04.038.

[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), 437-446. doi: 10.1016/j.anihpc.2009.11.016.

[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), 5832-5851. doi: 10.1016/j.jde.2012.01.045.

[4]

Y. S. Choi and Z. A. Wang, Prevention of blow-up by fast diffusion in chemotaxis, J. Math. Anal. Appl., 362 (2010), 553-564. doi: 10.1016/j.jmaa.2009.08.012.

[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), 137-149. doi: 10.1016/j.ydbio.2006.04.451.

[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), 831-844. doi: 10.1111/j.1460-9568.2004.03213.x.

[7]

T. Hillen and K. Painter, A users guide to PDE models for chemotaxis, J. Math.Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[8]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.

[9]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. Deutsch.Math.-Verien, 106 (2004), 51-69.

[10]

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.

[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), 2993-3010. doi: 10.1016/j.jde.2014.01.028.

[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), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569.

[13]

H. Y. Jin, Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478. doi: 10.1016/j.jmaa.2014.09.049.

[14]

H. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080.

[15]

H. Jin and Z. A. Wang, Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 260 (2016), 162-196. doi: 10.1016/j.jde.2015.08.040.

[16]

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.

[17]

X. Li and Z. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Continuous Dynam. Systems, 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.

[18]

X. Li and Z. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198. doi: 10.1093/imamat/hxv033.

[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), 31-41. doi: 10.1080/17513758.2011.571722.

[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), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.

[21]

D. Liu and Y. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546. doi: 10.1002/mma.3240.

[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), 1228-1262. doi: 10.1137/130934246.

[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), 693-730. doi: 10.1016/S0092-8240(03)00030-2.

[24]

T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.

[25]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa , 20 (1966), 733-737.

[26]

K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.

[27]

K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.

[28]

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.

[29]

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.

[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), 2520-2543. doi: 10.1016/j.jde.2011.07.010.

[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), 789-802. doi: 10.3934/dcds.2014.34.789.

[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), 1847-1872. doi: 10.1016/j.jde.2013.12.007.

[33]

Y. Wang and Z. Xiang, Global existence and boundedness in a higher-dimensional quasilinear chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 3159-3179. doi: 10.1007/s00033-015-0557-3.

[34]

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.

[35]

M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319.

[36]

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.

[37]

M. Winkler, Does a volume-filling effect always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24. doi: 10.1002/mma.1146.

[38]

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.

[39]

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.

[40]

M. Winkler and K. C. Djie, Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal., 72 (2010), 1044-1064. doi: 10.1016/j.na.2009.07.045.

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