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February 2019, 24(2): 423-447. doi: 10.3934/dcdsb.2018180

Global existence for an attraction-repulsion chemotaxis fluid model with logistic source

1. 

Universidad Industrial de Santander, Escuela de Matemáticas, Bucaramanga, A.A. 678, Colombia

2. 

Universidade Estadual de Campinas, Departamento de Matemática-IMECC, CEP 13083-859, Campinas-SP, Brazil

* Corresponding author: Élder J. Villamizar-Roa

Received  November 2017 Revised  January 2018 Published  June 2018

Fund Project: The second author has been partially supported by CNPq and FAPESP, Brazil. The third author has been supported by Vicerrectoría de Investigación y Extensión of Universidad Industrial de Santander, and Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación Francisco José de Caldas, contrato Colciencias FP 44842-157-2016. The authors would like to thank an anonymous referee for useful remarks and suggestions

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of $\mathbb{R}^N,$ $N = 2, 3,$ for small initial data in $L^p$-spaces.

Citation: Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
References:
[1]

M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Analysis: Real World Applications, 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011.

[2]

L. AngiuliD. Pallara and F. Y. Paronetto, Analytic semigroups generated in L1 by second order elliptic operators via duality methods, Semigroup Forum, Springer, 80 (2010), 255-271. doi: 10.1007/s00233-009-9200-y.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[4]

M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 34 (2017), 1013-1039. doi: 10.1016/j.anihpc.2016.08.003.

[5]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.

[6]

T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998), 83-120. doi: 10.1007/PL00004606.

[7]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.

[8]

M.A. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947.

[9]

M. A. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149.

[10]

A. ChertockK. FellnerA. KurganovA. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012), 155-190. doi: 10.1017/jfm.2011.534.

[11]

H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017), 1415-1426. doi: 10.1016/j.jmaa.2016.09.050.

[12]

C. DombrowskiL. CisnerosS. ChatkaewR. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004), 98-103.

[13]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012), 1833-1852. doi: 10.1093/imrn/rns270.

[14]

E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126. doi: 10.1016/j.nonrwa.2014.07.001.

[15]

L. C. F. Ferreira and E. J. Villamizar-Roa, Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370.

[16]

L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011.

[17]

D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977), 685-700.

[18]

Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981), 297-329. doi: 10.1007/BF01214869.

[19]

N. A. Hill and T. J. Pedley, Bioconvection, Fluid Dynamics Research, 37 (2005), 1-20. doi: 10.1016/j.fluiddyn.2005.03.002.

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T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[21]

T. HillenK. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198. doi: 10.1142/S0218202512500480.

[22]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.

[23]

S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463.

[24]

J. JiangH. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015), 249-258.

[25]

T. Kato, Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.

[26]

T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992), 127-155. doi: 10.1007/BF01232939.

[27]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879.

[28]

H. KozonoM. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016), 1663-1683. doi: 10.1016/j.jfa.2015.10.016.

[29]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016.

[30]

J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X.

[31]

D. LiC. MuK. Lin and L. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016), 914-936. doi: 10.1016/j.jmaa.2016.11.036.

[32]

X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 289-301. doi: 10.1002/mma.3477.

[33]

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

[34]

J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028.

[35]

J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017), 5271-5305. doi: 10.1016/j.jde.2017.01.024.

[36]

P. LiuJ. Shi and Z.-A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.

[37]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.

[38]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.

[39]

N. V. MantzarisS. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.

[40]

X. Mora, Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983), 21-55. doi: 10.2307/1999300.

[41]

A. Quinlan and B. Straughan, Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005), 2887-2897. doi: 10.1098/rspa.2005.1483.

[42]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.

[43]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y.

[44]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.

[45]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[46]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.

[47]

R. TysonS. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, Journal of Mathematical Biology, 38 (1999), 359-375. doi: 10.1007/s002850050153.

[48]

Y. Wang, Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Electronic Journal of Differential Equations, 176 (2016), 1-21.

[49]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, Journal of Differential Equations, 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010.

[50]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[51]

M. Winkler, Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Communications in Partial Differential Equations, 35 (2010), 1516-1537. doi: 10.1080/03605300903473426.

[52]

M. Winkler, Global large-data solutions in a chemotaxis-(Navier-) Stokes system modeling cellular swimming in fluid drops, Communications in Partial Differential Equations, 37 (2012), 319-351. doi: 10.1080/03605302.2011.591865.

[53]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, Journal of Differential Equations, 257 (2014), 1056-1077. doi: 10.1016/j.jde.2014.04.023.

[54]

M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calculus of Variations and Partial Differential Equations, 54 (2015), 3789-3828. doi: 10.1007/s00526-015-0922-2.

[55]

M. Winkler, Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 33 (2016), 1329-1352. doi: 10.1016/j.anihpc.2015.05.002.

[56]

D. Woodward, R. Tyson, M. Myerscough, J. D. Murray, E. Budrene, and H. Berg, Spatio-temporal patterns generated by Salmonella typhimurium Biophysical Journal, 68 (1995), no. 5, 2181. doi: 10.1016/S0006-3495(95)80400-5.

[57]

Q. Zhang and Y. Li, An attraction-repulsion chemotaxis system with logistic source, Biophysical Journal, 96 (2015), 570-584. doi: 10.1002/zamm.201400311.

[58]

Q. Zhang and Y. Li, Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, Journal of Differential Equations, 259 (2015), 3730-3754. doi: 10.1016/j.jde.2015.05.012.

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P. ZhengC. Mu and X. Hu, Boundedness in the higher dimensional attraction-repulsion chemotaxis-growth system, Computers & Mathematics with Applications, 72 (2016), 2194-2202. doi: 10.1016/j.camwa.2016.08.028.

show all references

References:
[1]

M. AidaK. OsakiT. TsujikawaA. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Analysis: Real World Applications, 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011.

[2]

L. AngiuliD. Pallara and F. Y. Paronetto, Analytic semigroups generated in L1 by second order elliptic operators via duality methods, Semigroup Forum, Springer, 80 (2010), 255-271. doi: 10.1007/s00233-009-9200-y.

[3]

N. BellomoA. BellouquidY. Tao and M. Winkler, Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Mathematical Models and Methods in Applied Sciences, 25 (2015), 1663-1763. doi: 10.1142/S021820251550044X.

[4]

M. Braukhoff, Global (weak) solution of the chemotaxis-Navier-Stokes equations with non-homogeneous boundary conditions and logistic growth, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 34 (2017), 1013-1039. doi: 10.1016/j.anihpc.2016.08.003.

[5]

X. Cao and J. Lankeit, Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities, Calc. Var. Partial Differential Equations, 55 (2016), Art. 107, 39 pp. doi: 10.1007/s00526-016-1027-2.

[6]

T. Cazenave and F. B. Weissler, Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations, Mathematische Zeitschrift, 228 (1998), 83-120. doi: 10.1007/PL00004606.

[7]

S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, The International Series of Monographs on Physics Clarendon Press, Oxford, 1961.

[8]

M.A. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of tissue: The role of the urokinase plasminogen activation system, Mathematical Models and Methods in Applied Sciences, 15 (2005), 1685-1734. doi: 10.1142/S0218202505000947.

[9]

M. A. Chaplain and A. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, Mathematical Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149.

[10]

A. ChertockK. FellnerA. KurganovA. Lorz and P. Markowich, Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach, Journal of Fluid Mechanics, 694 (2012), 155-190. doi: 10.1017/jfm.2011.534.

[11]

H. J. Choe and B. Lkhagvasuren, Global existence result for chemotaxis Navier-Stokes equations in the critical Besov spaces, Journal of Mathematical Analysis and Applications, 446 (2017), 1415-1426. doi: 10.1016/j.jmaa.2016.09.050.

[12]

C. DombrowskiL. CisnerosS. ChatkaewR. E. Goldstein and J. O. Kessler, Self-concentration and large-scale coherence in bacterial dynamics, Physical Review Letters, 93 (2004), 98-103.

[13]

R. Duan and Z. Xiang, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, International Mathematics Research Notices, 2014 (2012), 1833-1852. doi: 10.1093/imrn/rns270.

[14]

E. Espejo and T. Suzuki, Reaction terms avoiding aggregation in slow fluids, Nonlinear Analysis: Real World Applications, 21 (2015), 110-126. doi: 10.1016/j.nonrwa.2014.07.001.

[15]

L. C. F. Ferreira and E. J. Villamizar-Roa, Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations, Differential and Integral Equations, 19 (2006), 1349-1370.

[16]

L. C. F. Ferreira and E. J. Villamizar-Roa, Well-posedness and asymptotic behaviour for the convection problem, Nonlinearity, 19 (2006), 2169-2191. doi: 10.1088/0951-7715/19/9/011.

[17]

D. Fujiwara and H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, IA Math, 24 (1977), 685-700.

[18]

Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Mathematische Zeitschrift, 178 (1981), 297-329. doi: 10.1007/BF01214869.

[19]

N. A. Hill and T. J. Pedley, Bioconvection, Fluid Dynamics Research, 37 (2005), 1-20. doi: 10.1016/j.fluiddyn.2005.03.002.

[20]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[21]

T. HillenK. J. Painter and M. Winkler, Convergence of a cancer invasion model to a logistic chemotaxis model, Mathematical Models and Methods in Applied Sciences, 23 (2013), 165-198. doi: 10.1142/S0218202512500480.

[22]

D. Horstmann, Generalizing the Keller-Segel model: Lyapunov functionals, steady state analysis, and blow-up results for multi-species chemotaxis models in the presence of attraction and repulsion between competitive interacting species, Journal of Nonlinear Science, 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.

[23]

S. Ishida, Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains, Discrete & Continuous Dynamical Systems-A, 35 (2015), 3463-3482. doi: 10.3934/dcds.2015.35.3463.

[24]

J. JiangH. Wu and S. Zheng, Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains, Asymptotic Analysis, 92 (2015), 249-258.

[25]

T. Kato, Strong Lp-solutions of the Navier-Stokes equation in Rm with applications to weak solutions, Math. Z., 187 (1984), 471-480. doi: 10.1007/BF01174182.

[26]

T. Kato, Strong solutions of the Navier-Stokes equation in Morrey spaces, Bol. Soc. Brasil. Mat., 22 (1992), 127-155. doi: 10.1007/BF01232939.

[27]

A. Kiselev and L. Ryzhik, Biomixing by chemotaxis and enhancement of biological reactions, Communications in Partial Differential Equations, 37 (2012), 298-318. doi: 10.1080/03605302.2011.589879.

[28]

H. KozonoM. Miura and Y. Sugiyama, Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, Journal of Functional Analysis, 270 (2016), 1663-1683. doi: 10.1016/j.jfa.2015.10.016.

[29]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, Journal of Differential Equations, 258 (2015), 1158-1191. doi: 10.1016/j.jde.2014.10.016.

[30]

J. Lankeit, Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109. doi: 10.1142/S021820251640008X.

[31]

D. LiC. MuK. Lin and L. Wang, Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, Journal of Mathematical Analysis and Applications, 448 (2016), 914-936. doi: 10.1016/j.jmaa.2016.11.036.

[32]

X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Mathematical Methods in the Applied Sciences, 39 (2016), 289-301. doi: 10.1002/mma.3477.

[33]

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

[34]

J. Liu and Y. Wang, Global existence and boundedness in a Keller-Segel-(Navier-) Stokes system with signal-dependent sensitivity, Journal of Mathematical Analysis and Applications, 447 (2017), 499-528. doi: 10.1016/j.jmaa.2016.10.028.

[35]

J. Liu and Y. Wang, Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, Journal of Differential Equations, 262 (2017), 5271-5305. doi: 10.1016/j.jde.2017.01.024.

[36]

P. LiuJ. Shi and Z.-A. Wang, Pattern formation of the attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2597-2625. doi: 10.3934/dcdsb.2013.18.2597.

[37]

M. LucaA. Chavez-RossL. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, Bulletin of Mathematical Biology, 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.

[38]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.

[39]

N. V. MantzarisS. Webb and H. G. Othmer, Mathematical modeling of tumor-induced angiogenesis, Journal of Mathematical Biology, 49 (2004), 111-187. doi: 10.1007/s00285-003-0262-2.

[40]

X. Mora, Semilinear parabolic problems define semiflows on Ck spaces, Transactions of the American Mathematical Society, 278 (1983), 21-55. doi: 10.2307/1999300.

[41]

A. Quinlan and B. Straughan, Decay bounds in a model for aggregation of microglia: Application to Alzheimer's disease senile plaques, Proceedings of the Royal Society A, 461 (2005), 2887-2897. doi: 10.1098/rspa.2005.1483.

[42]

Y. Tao and M. Winkler, Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 30 (2013), 157-178. doi: 10.1016/j.anihpc.2012.07.002.

[43]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Zeitschrift für angewandte Mathematik und Physik, 66 (2015), 2555-2573. doi: 10.1007/s00033-015-0541-y.

[44]

Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp. doi: 10.1007/s00033-016-0732-1.

[45]

J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Communications in Partial Differential Equations, 32 (2007), 849-877. doi: 10.1080/03605300701319003.

[46]

I. TuvalL. CisnerosC. DombrowskiC. W. WolgemuthJ. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proceedings of the National Academy of Sciences of the United States of America, 102 (2005), 2277-2282. doi: 10.1073/pnas.0406724102.

[47]

R. TysonS. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, Journal of Mathematical Biology, 38 (1999), 359-375. doi: 10.1007/s002850050153.

[48]

Y. Wang, Boundedness in a three-dimensional attraction-repulsion chemotaxis system with nonlinear diffusion and logistic source, Electronic Journal of Differential Equations, 176 (2016), 1-21.

[49]

Y. Wang and Z. Xiang, Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, Journal of Differential Equations, 261 (2016), 4944-4973. doi: 10.1016/j.jde.2016.07.010.

[50]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

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