American Institute of Mathematical Sciences

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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 & 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.  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, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
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M. A. Herrero and J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann., 306 (1996), 583-623.  doi: 10.1007/BF01445268.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Soc., (1988).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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, Comm. 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, Comm. Pure Appl. Math., 17 (1964), 35-92.  doi: 10.1002/cpa.3160170104.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Soc., (1988).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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