American Institute of Mathematical Sciences

Advanced
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

Hai-Yang Jin 1, and  Tian Xiang 2,,

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.
Keywords: Chemotaxis, attraction-repulsion, repulsion effects, boundedness, blow-up.
Mathematics Subject Classification: Primary: 35K51, 35K59, 35B45, 35A01; Secondary: 35Q92, 92C17.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Friedman, Partial Differential Equations Holt, Rinehart Winston, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[30]

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

[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.   Google Scholar

[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.  Google Scholar

[33]

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

[34]

K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Friedman, Partial Differential Equations Holt, Rinehart Winston, New York, 1969.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[30]

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

[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.   Google Scholar

[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.  Google Scholar

[33]

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

[34]

K. J. Painter and T. Hillen, Volume-filling quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[40]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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