# American Institute of Mathematical Sciences

October  2018, 23(8): 3071-3085. doi: 10.3934/dcdsb.2017197

## Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

 1 Department of Mathematics, South China University of Technology, Guangzhou 510640, China 2 Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author

Received  March 2017 Published  July 2017

Fund Project: The research of H.Y. Jin was supported by NSF of China (No. 11501218) and the Fundamental Research Funds for the Central Universities (No. 2017MS107), and the research of T. Xiang was supported by the NSF of China (No. 11601516,11571364 and 11571363)

We consider the following attraction-repulsion Keller-Segel system:
 $\begin{equation*}\begin{cases}u_t=\nabla· (D(u) \nabla u)-χ\nabla·( u\nabla v)+ξ\nabla·( u\nabla w), &x∈ Ω, ~~t>0,\\ v_t=Δ v+α u-β v,& x∈ Ω, ~~t>0,\\0=Δ w+γ u-δ w, &x∈ Ω, ~~t>0,\\u(x,0)=u_0(x),~v(x,0)=v_0(x),&x∈ Ω,\end{cases}\end{equation*}$
with homogeneous Neumann boundary conditions in a bounded domain $Ω\subset \mathbb{R}^n(n>2)$ with smooth boundary. Here all the parameters
 $χ, ξ, α, β, γ$
and
 $δ$
are positive. The smooth diffusion
 $D(u)$
satisfies
 $D(u)≥ d u^θ, u>0$
for some
 $d>0, θ∈\mathbb{R}$
. It is recently known from [25] that boundedness of solutions is ensured whenever
 $θ>1-\frac{2}{n}$
. Here, it is shown, if repulsion dominates or cancels attraction in the sense either
 $\{ξγ> χα\}$
or
 $\{ξγ=χα, β≥ δ\}$
, the corresponding initial-boundary value problem possesses a unique global classical solution which is uniformly-in-time bounded for large initial data provided
 $θ>1-\frac{4}{n+2}$
. In this way, the range of
 $θ>1-\frac{2}{n}$
of boundedness is enlarged and thus the repulsion effect on boundedness is exhibited.
Citation: Hai-Yang Jin, Tian Xiang. Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3071-3085. doi: 10.3934/dcdsb.2017197
##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104. [3] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. [4] N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [5] A. Blanchet, J. A. Carrillo and Ph. Laurencǫt, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7. [6] T. Cieślak, Ph. Laurencǫt and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, equations, Parabolic and Navier-Stokes Equations, in: Banach Center Publ. Polish Acad. Sci. Inst. Math., 81 (2008), 105-117. doi: 10.4064/bc81-0-7. [7] 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. [8] T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004. [9] E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. [10] A. Friedman, Partial Differential Equations Holt, Rinehart Winston, New York, 1969. [11] K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151. [12] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165. [13] 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), 745-760. doi: 10.1002/mma.2622. [14] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028. [15] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012. [16] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate KellerSegel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569. [17] K. Ishige, Ph. Laurençot and N. Mizoguchi, Blow-up behavior of solutions to a degenerate parabolic-parabolic Keller-Segel system, Math. Ann., 367 (2017), 461-499. doi: 10.1007/s00208-016-1400-7. [18] 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. [19] H. Y. Jin and Z. Liu, Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20. doi: 10.1016/j.aml.2015.03.004. [20] H. Y. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080. [21] H. Y. 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. [22] P. Laurençot and N. Mizoguchi, Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 197-220. doi: 10.1016/j.anihpc.2015.11.002. [23] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183. doi: 10.1016/j.nonrwa.2015.12.003. [24] K. Lin and C. Mu, Global existence and convergence to steady states for an attractionrepulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642. doi: 10.1016/j.nonrwa.2016.03.012. [25] K. Lin, C. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, J. Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002. [26] D. Liu and Y. S. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546. doi: 10.1002/mma.3240. [27] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn. Suppl.1, 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722. [28] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst.-Series B., 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [29] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and Alzheimer's disease senile plagues: is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. [30] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math.Sci. Appl., 5 (1995), 581-601. [31] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. Ser. Internat., 40 (1997), 411-433. [32] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042. [33] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737. [34] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [35] R. Shi and W. Wang, Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520. doi: 10.1016/j.jmaa.2014.10.006. [36] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003. [37] Y. S. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722. doi: 10.3934/dcdsb.2013.18.2705. [38] Y. S. 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. [39] Y. S. 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. [40] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319. [41] 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. [42] 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. [43] H. Yu, Q. Guo and S. Zheng, Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342. doi: 10.1016/j.nonrwa.2016.09.007.

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##### References:
 [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Commun. Pure Appl. Math., 12 (1959), 623-727. doi: 10.1002/cpa.3160120405. [2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Ⅱ, Commun. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104. [3] N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113. [4] N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X. [5] A. Blanchet, J. A. Carrillo and Ph. Laurencǫt, Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions, Calc. Var. Partial Differential Equations, 35 (2009), 133-168. doi: 10.1007/s00526-008-0200-7. [6] T. Cieślak, Ph. Laurencǫt and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, equations, Parabolic and Navier-Stokes Equations, in: Banach Center Publ. Polish Acad. Sci. Inst. Math., 81 (2008), 105-117. doi: 10.4064/bc81-0-7. [7] 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. [8] T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113. doi: 10.1016/j.jde.2014.12.004. [9] E. Espejo and T. Suzuki, Global existence and blow-up for a system describing the aggregation of microglia, Appl. Math. Lett., 35 (2014), 29-34. doi: 10.1016/j.aml.2014.04.007. [10] A. Friedman, Partial Differential Equations Holt, Rinehart Winston, New York, 1969. [11] K. Fujie, A. Ito, M. Winkler and T. Yokota, Stabilization in a chemotaxis model for tumor invasion, Discrete Contin. Dyn. Syst., 36 (2016), 151-169. doi: 10.3934/dcds.2016.36.151. [12] D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Deutsch. Math. Verien., 105 (2003), 103-165. [13] 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), 745-760. doi: 10.1002/mma.2622. [14] S. Ishida, K. Seki and T. Yokota, Boundedness in quasilinear Keller-Segel systems of parabolicparabolic type on non-convex bounded domains, J. Differential Equations, 256 (2014), 2993-3010. doi: 10.1016/j.jde.2014.01.028. [15] S. Ishida and T. Yokota, Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type, J. Differential Equations, 252 (2012), 1421-1440. doi: 10.1016/j.jde.2011.02.012. [16] S. Ishida and T. Yokota, Blow-up in finite or infinite time for quasilinear degenerate KellerSegel systems of parabolic-parabolic type, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2569-2596. doi: 10.3934/dcdsb.2013.18.2569. [17] K. Ishige, Ph. Laurençot and N. Mizoguchi, Blow-up behavior of solutions to a degenerate parabolic-parabolic Keller-Segel system, Math. Ann., 367 (2017), 461-499. doi: 10.1007/s00208-016-1400-7. [18] 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. [19] H. Y. Jin and Z. Liu, Large time behavior of the full attraction-repulsion Keller-Segel system in the whole space, Appl. Math. Lett., 47 (2015), 13-20. doi: 10.1016/j.aml.2015.03.004. [20] H. Y. Jin and Z. A. Wang, Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Methods Appl. Sci., 38 (2015), 444-457. doi: 10.1002/mma.3080. [21] H. Y. 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. [22] P. Laurençot and N. Mizoguchi, Finite time blowup for the parabolic-parabolic Keller-Segel system with critical diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 197-220. doi: 10.1016/j.anihpc.2015.11.002. [23] Y. Li and Y. X. Li, Blow-up of nonradial solutions to attraction-repulsion chemotaxis system in two dimensions, Nonlinear Anal. Real Word Appl., 30 (2016), 170-183. doi: 10.1016/j.nonrwa.2015.12.003. [24] K. Lin and C. Mu, Global existence and convergence to steady states for an attractionrepulsion chemotaxis system, Nonlinear Anal. Real World Appl., 31 (2016), 630-642. doi: 10.1016/j.nonrwa.2016.03.012. [25] K. Lin, C. Mu and Y. Gao, Boundedness and blow up in the higher-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion, J. Differential Equations, 261 (2016), 4524-4572. doi: 10.1016/j.jde.2016.07.002. [26] D. Liu and Y. S. Tao, Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546. doi: 10.1002/mma.3240. [27] J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn. Suppl.1, 6 (2012), 31-41. doi: 10.1080/17513758.2011.571722. [28] P. Liu, J. P. Shi and Z. A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst.-Series B., 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597. [29] M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, Microglia, and Alzheimer's disease senile plagues: is there a connection?, Bull. Math. Biol., 65 (2003), 693-730. [30] T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math.Sci. Appl., 5 (1995), 581-601. [31] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac. Ser. Internat., 40 (1997), 411-433. [32] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042. [33] L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 20 (1966), 733-737. [34] K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. [35] R. Shi and W. Wang, Well-posedness for a model derived from an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 423 (2015), 497-520. doi: 10.1016/j.jmaa.2014.10.006. [36] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Differential Equations, 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003. [37] Y. S. Tao, Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2705-2722. doi: 10.3934/dcdsb.2013.18.2705. [38] Y. S. 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. [39] Y. S. 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. [40] M. Winkler, A critical exponent in a degenerate parabolic equation, Math. Methods Appl. Sci., 25 (2002), 911-925. doi: 10.1002/mma.319. [41] 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. [42] 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. [43] H. Yu, Q. Guo and S. Zheng, Finite time blow-up of nonradial solutions in an attraction-repulsion chemotaxis system, Nonlinear Anal. Real World Appl., 34 (2017), 335-342. doi: 10.1016/j.nonrwa.2016.09.007.
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