Global existence of solutions to an attraction-repulsion chemotaxis model with growth

Sainan Wu; Junping Shi; Boying Wu

doi:10.3934/cpaa.2017050

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May 2017, 16(3): 1037-1058. doi: 10.3934/cpaa.2017050

Global existence of solutions to an attraction-repulsion chemotaxis model with growth

Sainan Wu ^1, , Junping Shi ^2, and Boying Wu ^1,

1.	Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China
2.	Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Received June 2016 Revised November 2016 Published February 2017

Fund Project: S. Wu is supported in part by a grant from China Scholarship Council; J. Shi is partially supported by US-NSF grant DMS-1313243 and B. Wu is partially supported by National Natural Science Foundation of China grant No. 11271100.

An attraction-repulsion chemotaxis model with nonlinear chemotactic sensitivity functions and growth source is considered. The global-in-time existence and boundedness of solutions are proved under some conditions on the nonlinear sensitivity functions and growth source function. Our results improve the earlier ones for the linear sensitivity functions.

Keywords: Attraction-repulsion chemotaxis, global existence, boundedness, nonlinear chemotactic sensitivity.

Mathematics Subject Classification: 35K57, 35K59; 35B45, 92C17.

Citation: Sainan Wu, Junping Shi, Boying Wu. Global existence of solutions to an attraction-repulsion chemotaxis model with growth. Communications on Pure and Applied Analysis, 2017, 16 (3) : 1037-1058. doi: 10.3934/cpaa.2017050

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, Comm. 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, Comm. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104.
[3]	M. Aida, K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011.
[4]	N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Towards 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]	T. Ciéslak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Banach Center Publ, 81 (2008), 105-117. doi: 10.4064/bc81-0-7.
[6]	T. Ciéslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.
[7]	K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684. doi: 10.1016/j.jmaa.2014.11.045.
[8]	K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic KellerSegel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224. doi: 10.1002/mma.3149.
[9]	K. Fujie and T. Yokota, Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity, Appl. Math. Lett., 38 (2014), 140-143. doi: 10.1016/j.aml.2014.07.021.
[10]	M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gates and S. B. Dunnett, Spatially and temporally restricted chemoattractive and chemorepulsive cues direct the formation of the nigro-striatal circuit, Eur. J. Neurosci., 19 (2004), 831-844. doi: 10.1111/j.1460-9568.2004.03213.x.
[11]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.
[12]	M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.
[13]	D. Horstmann and G. F. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[14]	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.
[15]	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.
[16]	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.
[17]	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.
[18]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1002/mma.3080.
[19]	O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Soc., (1988).
[20]	X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301. doi: 10.1002/mma.3477.
[21]	X. Li and Z. Y. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.
[22]	X. Li and Z. Y. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198. doi: 10.1093/imamat/hxv033.
[23]	J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 1751-3758. doi: 10.1080/17513758.2011.571722.
[24]	D. M. 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.
[25]	P. Liu, J. P. 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.
[26]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, B. Math. Biol., 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.
[27]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
[28]	L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. doi: 10.1007/978-3-642-10926-3_1.
[29]	K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
[30]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. doi: 10.1.1.641.4757.
[31]	B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012.
[32]	R. K. Shi and W. K. 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.
[33]	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.
[34]	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.
[35]	Y. S. Tao and M. Winkler, A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.
[36]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial. Diff. Eqns., 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[37]	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.
[38]	Z. A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Commun. Pure Appl. Anal., 12 (2013), 3027-3046. doi: 10.3934/cpaa.2013.12.3027.
[39]	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.
[40]	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.
[41]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[42]	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.
[43]	X. S. Yang, D. Dormann, A. E. Münsterberg and C. J. Weijer, Cell movement patterns during gastrulation in the chick are controlled by positive and negative chemotaxis mediated by FGF4 and FGF8, Dev. Cell, 3 (2002), 425-437. doi: 10.1016/S1534-5807(02)00256-3.
[44]	Q. S. Zhang and Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, Z. Angew. Math. Mech., 96 (2016), 570-584. doi: 10.1002/zamm.201400311.
[45]	J. S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
[46]	P. Zheng, C. L. Mu, X. G. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031.
[47]	P. Zheng, C. L. Mu, X. G. Hu and Q. H. Zhang, Global boundedness in a quasilinear chemotaxis system with signal-dependent sensitivity, J. Math. Anal. Appl., 428 (2015), 508-524. doi: 10.1016/j.jmaa.2015.03.047.

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, Comm. 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, Comm. Pure Appl. Math., 17 (1964), 35-92. doi: 10.1002/cpa.3160170104.
[3]	M. Aida, K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura, Chemotaxis and growth system with singular sensitivity function, Nonlinear Anal. Real World Appl., 6 (2005), 323-336. doi: 10.1016/j.nonrwa.2004.08.011.
[4]	N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler, Towards 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]	T. Ciéslak, P. Laurençot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system, Banach Center Publ, 81 (2008), 105-117. doi: 10.4064/bc81-0-7.
[6]	T. Ciéslak and M. Winkler, Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21 (2008), 1057-1076. doi: 10.1088/0951-7715/21/5/009.
[7]	K. Fujie, Boundedness in a fully parabolic chemotaxis system with singular sensitivity, J. Math. Anal. Appl., 424 (2015), 675-684. doi: 10.1016/j.jmaa.2014.11.045.
[8]	K. Fujie, M. Winkler and T. Yokota, Boundedness of solutions to parabolic-elliptic KellerSegel systems with signal-dependent sensitivity, Math. Methods Appl. Sci., 38 (2015), 1212-1224. doi: 10.1002/mma.3149.
[9]	K. Fujie and T. Yokota, Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity, Appl. Math. Lett., 38 (2014), 140-143. doi: 10.1016/j.aml.2014.07.021.
[10]	M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gates and S. B. Dunnett, Spatially and temporally restricted chemoattractive and chemorepulsive cues direct the formation of the nigro-striatal circuit, Eur. J. Neurosci., 19 (2004), 831-844. doi: 10.1111/j.1460-9568.2004.03213.x.
[11]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.
[12]	M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623. doi: 10.1007/BF01445268.
[13]	D. Horstmann and G. F. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177. doi: 10.1017/S0956792501004363.
[14]	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.
[15]	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.
[16]	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.
[17]	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.
[18]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415. doi: 10.1002/mma.3080.
[19]	O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Soc., (1988).
[20]	X. Li, Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301. doi: 10.1002/mma.3477.
[21]	X. Li and Z. Y. Xiang, Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 3503-3531. doi: 10.3934/dcds.2015.35.3503.
[22]	X. Li and Z. Y. Xiang, On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198. doi: 10.1093/imamat/hxv033.
[23]	J. Liu and Z. A. Wang, Classical solutions and steady states of an attraction-repulsion chemotaxis in one dimension, J. Biol. Dyn., 6 (2012), 1751-3758. doi: 10.1080/17513758.2011.571722.
[24]	D. M. 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.
[25]	P. Liu, J. P. 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.
[26]	M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signaling, microglia, and Alzheimer's disease senile plaques: Is there a connection?, B. Math. Biol., 65 (2003), 693-730. doi: 10.1016/S0092-8240(03)00030-2.
[27]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
[28]	L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3), 13 (1959), 115-162. doi: 10.1007/978-3-642-10926-3_1.
[29]	K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
[30]	K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543. doi: 10.1.1.641.4757.
[31]	B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270. doi: 10.1088/0951-7715/24/4/012.
[32]	R. K. Shi and W. K. 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.
[33]	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.
[34]	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.
[35]	Y. S. Tao and M. Winkler, A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704. doi: 10.1137/100802943.
[36]	J. I. Tello and M. Winkler, A chemotaxis system with logistic source, Commun. Partial. Diff. Eqns., 32 (2007), 849-877. doi: 10.1080/03605300701319003.
[37]	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.
[38]	Z. A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Commun. Pure Appl. Anal., 12 (2013), 3027-3046. doi: 10.3934/cpaa.2013.12.3027.
[39]	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.
[40]	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.
[41]	M. Winkler, Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272. doi: 10.1016/j.jmaa.2011.05.057.
[42]	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.
[43]	X. S. Yang, D. Dormann, A. E. Münsterberg and C. J. Weijer, Cell movement patterns during gastrulation in the chick are controlled by positive and negative chemotaxis mediated by FGF4 and FGF8, Dev. Cell, 3 (2002), 425-437. doi: 10.1016/S1534-5807(02)00256-3.
[44]	Q. S. Zhang and Y. X. Li, An attraction-repulsion chemotaxis system with logistic source, Z. Angew. Math. Mech., 96 (2016), 570-584. doi: 10.1002/zamm.201400311.
[45]	J. S. Zheng, Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic source, J. Differential Equations, 259 (2015), 120-140. doi: 10.1016/j.jde.2015.02.003.
[46]	P. Zheng, C. L. Mu, X. G. Hu and Y. Tian, Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source, J. Math. Anal. Appl., 424 (2015), 509-522. doi: 10.1016/j.jmaa.2014.11.031.
[47]	P. Zheng, C. L. Mu, X. G. Hu and Q. H. Zhang, Global boundedness in a quasilinear chemotaxis system with signal-dependent sensitivity, J. Math. Anal. Appl., 428 (2015), 508-524. doi: 10.1016/j.jmaa.2015.03.047.

Figure 1. Regions in (p, q) plane where the global existence and boundedness of solutions to (1.9) are proved. The regions labelled by (ⅰ), (ⅱ), (ⅲ) and (iv) correspond to the ones defined in Theorems 1.1 and 1.2, and for the region labelled with?, the result is not known. Left: n ≤ 2; Right: n > 2

1. $n = 1, 2$, and $0\le p, q\le 2/n$, or $2/n\le \max\{p, q\}\le r-1$ and $b$ large; or
2. $n\ge 3$, and $0\le p, q\le 2/n$, or $1\le \max\{p, q\}\le r-1$ and $b$ large.

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2020 Impact Factor: 1.916

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2020 Impact Factor: 1.916

American Institute of Mathematical Sciences

Global existence of solutions to an attraction-repulsion chemotaxis model with growth

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American Institute of Mathematical Sciences

Global existence of solutions to an attraction-repulsion chemotaxis model with growth

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