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On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation
Boundedness in a two species attraction-repulsion chemotaxis system with two chemicals
1. | School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China |
2. | School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China |
3. | Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada |
This paper deals with a class of attraction-repulsion chemotaxis systems in a smoothly bounded domain. When the system is parabolic-elliptic-parabolic-elliptic and the domain is $ n $-dimensional, if the repulsion effect is strong enough then the solutions of the system are globally bounded. Meanwhile, when the system is fully parabolic and the domain is either one-dimensional or two-dimensional, the system also possesses a globally bounded classical solution.
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. Bellomo, A. Bellouquid, Y. 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. |
[4] |
P. Biler, E. E. Espejo and I. Guerra,
Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.
doi: 10.3934/cpaa.2013.12.89. |
[5] |
P. Biler, W. Hebisch and T. Nadzieja,
The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[7] |
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. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[9] |
Q. Guo, Z. Jiang and S. Zheng,
Critical mass for an attraction-repulsion chemotaxis system, Appl. Anal., 97 (2018), 2349-2354.
doi: 10.1080/00036811.2017.1366989. |
[10] |
X. He, M. Tian and S. Zheng, Large time behavior of solutions to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 54 (2020), 103095, 14 pp.
doi: 10.1016/j.nonrwa.2020.103095. |
[11] |
M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson,
Bacterial competition: Surviving and thriving in the microbial jungle, Nat. Rev. Microbiol., 8 (2010), 15-25.
doi: 10.1038/nrmicro2259. |
[12] |
H.-Y. Jin and Z.-A. Wang,
Global stabilization of the full attraction-repulsion Kesser-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509-3527.
doi: 10.3934/dcds.2020027. |
[13] |
H.-Y. Jin and T. Xiang,
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimenssions, Discrete Contin. Dyn. Syst. B., 23 (2018), 3071-3085.
doi: 10.3934/dcdsb.2017197. |
[14] |
D. Li, C. Mu, K. Lin and L. Wang,
Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, J. Math. Anal. Appl., 448 (2017), 914-936.
doi: 10.1016/j.jmaa.2016.11.036. |
[15] |
J. Li, Y. Ke and Y. Wang,
Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 39 (2018), 261-277.
doi: 10.1016/j.nonrwa.2017.07.002. |
[16] |
J. Li and Y. Wang,
Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 467 (2018), 1066-1079.
doi: 10.1016/j.jmaa.2018.07.051. |
[17] |
X. Li and Y. Wang,
Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. B., 22 (2017), 2717-2729.
doi: 10.3934/dcdsb.2017132. |
[18] |
X. Li and Z. Xiang,
On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.
doi: 10.1093/imamat/hxv033. |
[19] |
Y. Li and W. Wang,
Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.
doi: 10.1002/mma.4942. |
[20] |
D. Liu and Y. Tao,
Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[21] |
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.
doi: 10.1016/S0092-8240(03)00030-2. |
[22] |
N. Mizoguchi and Ph. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[23] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
doi: 10.1007/b98869. |
[24] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[25] |
K. J. Painter,
Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.
doi: 10.1007/s11538-009-9396-8. |
[26] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Q., 10 (2002), 501-543.
|
[27] |
K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theoret. Biol., 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[28] |
H. Qiu and S. Guo,
Global existence and stablity in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. B., 24 (2019), 1569-1587.
doi: 10.3934/dcdsb.2018220. |
[29] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[30] |
Y. 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. |
[31] |
Y. Tao and M. Winkler,
Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.
doi: 10.1142/S021820251950043X. |
[32] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[33] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[34] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[35] |
M. Tian, X. He and S. Zheng,
Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 30 (2016), 1-15.
doi: 10.1016/j.nonrwa.2015.11.004. |
[36] |
X. Tu, C. Mu, P. Zheng and K. Lin,
Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.
doi: 10.3934/dcds.2018156. |
[37] |
G. Viglialoro,
Explicit lower bound of blow-up time for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 479 (2019), 1069-1077.
doi: 10.1016/j.jmaa.2019.06.067. |
[38] |
L. Wang and C. Mu,
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 4585-4601.
doi: 10.3934/dcdsb.2020114. |
[39] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stablization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[40] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[41] |
P. Xu and S. Zheng,
Global boundedness in an attraction-repulsion chemotaxis system with logistic source, Appl. Math. Lett., 83 (2018), 1-6.
doi: 10.1016/j.aml.2018.03.007. |
[42] |
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. |
[43] |
H. Yu, W. Wang and S. Zheng,
Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.
doi: 10.1088/1361-6544/aa96c9. |
[44] |
Y. Zeng,
Existence of global bounded classical solution to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal., 161 (2017), 182-197.
doi: 10.1016/j.na.2017.06.003. |
[45] |
Q. Zhang,
Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.
doi: 10.1016/j.aml.2018.03.012. |
[46] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
[47] |
Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9 pp.
doi: 10.1063/1.5011725. |
[48] |
J. Zhao, C. Mu, D. Zhou and K. Lin,
A parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with logistic source, J. Math. Anal. Appl., 455 (2017), 650-679.
doi: 10.1016/j.jmaa.2017.05.068. |
[49] |
J. Zheng,
Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.
doi: 10.12775/TMNA.2016.082. |
[50] |
P. Zheng and C. Mu,
Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.
doi: 10.1007/s10440-016-0083-0. |
[51] |
P. Zheng, C. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals, J. Math. Phys., 58 (2017), 111501, 17 pp.
doi: 10.1063/1.5010681. |
[52] |
P. Zheng, C. Mu and Y. Mi,
Global stability in a two-competing-species chemotaxis system with two chemicals, Differential Integral Equations, 31 (2018), 547-558.
|
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. |
[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. Bellomo, A. Bellouquid, Y. 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. |
[4] |
P. Biler, E. E. Espejo and I. Guerra,
Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.
doi: 10.3934/cpaa.2013.12.89. |
[5] |
P. Biler, W. Hebisch and T. Nadzieja,
The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. |
[7] |
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. |
[8] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[9] |
Q. Guo, Z. Jiang and S. Zheng,
Critical mass for an attraction-repulsion chemotaxis system, Appl. Anal., 97 (2018), 2349-2354.
doi: 10.1080/00036811.2017.1366989. |
[10] |
X. He, M. Tian and S. Zheng, Large time behavior of solutions to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 54 (2020), 103095, 14 pp.
doi: 10.1016/j.nonrwa.2020.103095. |
[11] |
M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson,
Bacterial competition: Surviving and thriving in the microbial jungle, Nat. Rev. Microbiol., 8 (2010), 15-25.
doi: 10.1038/nrmicro2259. |
[12] |
H.-Y. Jin and Z.-A. Wang,
Global stabilization of the full attraction-repulsion Kesser-Segel system, Discrete Contin. Dyn. Syst., 40 (2020), 3509-3527.
doi: 10.3934/dcds.2020027. |
[13] |
H.-Y. Jin and T. Xiang,
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimenssions, Discrete Contin. Dyn. Syst. B., 23 (2018), 3071-3085.
doi: 10.3934/dcdsb.2017197. |
[14] |
D. Li, C. Mu, K. Lin and L. Wang,
Large time behavior of solution to an attraction-repulsion chemotaxis system with logistic source in three dimensions, J. Math. Anal. Appl., 448 (2017), 914-936.
doi: 10.1016/j.jmaa.2016.11.036. |
[15] |
J. Li, Y. Ke and Y. Wang,
Large time behavior of solutions to a fully parabolic attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 39 (2018), 261-277.
doi: 10.1016/j.nonrwa.2017.07.002. |
[16] |
J. Li and Y. Wang,
Repulsion effects on boundedness in the higher dimensional fully parabolic attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 467 (2018), 1066-1079.
doi: 10.1016/j.jmaa.2018.07.051. |
[17] |
X. Li and Y. Wang,
Boundedness in a two-species chemotaxis parabolic system with two chemicals, Discrete Contin. Dyn. Syst. B., 22 (2017), 2717-2729.
doi: 10.3934/dcdsb.2017132. |
[18] |
X. Li and Z. Xiang,
On an attraction-repulsion chemotaxis system with a logistic source, IMA J. Appl. Math., 81 (2016), 165-198.
doi: 10.1093/imamat/hxv033. |
[19] |
Y. Li and W. Wang,
Boundedness in a four-dimensional attraction-repulsion chemotaxis system with logistic source, Math. Methods Appl. Sci., 41 (2018), 4936-4942.
doi: 10.1002/mma.4942. |
[20] |
D. Liu and Y. Tao,
Global boundedness in a fully parabolic attraction-repulsion chemotaxis model, Math. Methods Appl. Sci., 38 (2015), 2537-2546.
doi: 10.1002/mma.3240. |
[21] |
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.
doi: 10.1016/S0092-8240(03)00030-2. |
[22] |
N. Mizoguchi and Ph. Souplet,
Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[23] |
J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
doi: 10.1007/b98869. |
[24] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[25] |
K. J. Painter,
Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.
doi: 10.1007/s11538-009-9396-8. |
[26] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Q., 10 (2002), 501-543.
|
[27] |
K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theoret. Biol., 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[28] |
H. Qiu and S. Guo,
Global existence and stablity in a two-species chemotaxis system, Discrete Contin. Dyn. Syst. B., 24 (2019), 1569-1587.
doi: 10.3934/dcdsb.2018220. |
[29] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[30] |
Y. 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. |
[31] |
Y. Tao and M. Winkler,
Large time behavior in a forager-exploiter model with different taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.
doi: 10.1142/S021820251950043X. |
[32] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[33] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[34] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[35] |
M. Tian, X. He and S. Zheng,
Global boundedness in quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal. Real World Appl., 30 (2016), 1-15.
doi: 10.1016/j.nonrwa.2015.11.004. |
[36] |
X. Tu, C. Mu, P. Zheng and K. Lin,
Global dynamics in a two-species chemotaxis-competition system with two signals, Discrete Contin. Dyn. Syst., 38 (2018), 3617-3636.
doi: 10.3934/dcds.2018156. |
[37] |
G. Viglialoro,
Explicit lower bound of blow-up time for an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 479 (2019), 1069-1077.
doi: 10.1016/j.jmaa.2019.06.067. |
[38] |
L. Wang and C. Mu,
A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 4585-4601.
doi: 10.3934/dcdsb.2020114. |
[39] |
L. Wang, J. Zhang, C. Mu and X. Hu,
Boundedness and stablization in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. B., 25 (2020), 191-221.
doi: 10.3934/dcdsb.2019178. |
[40] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differ. Equ., 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[41] |
P. Xu and S. Zheng,
Global boundedness in an attraction-repulsion chemotaxis system with logistic source, Appl. Math. Lett., 83 (2018), 1-6.
doi: 10.1016/j.aml.2018.03.007. |
[42] |
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. |
[43] |
H. Yu, W. Wang and S. Zheng,
Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity, 31 (2018), 502-514.
doi: 10.1088/1361-6544/aa96c9. |
[44] |
Y. Zeng,
Existence of global bounded classical solution to a quasilinear attraction-repulsion chemotaxis system with logistic source, Nonlinear Anal., 161 (2017), 182-197.
doi: 10.1016/j.na.2017.06.003. |
[45] |
Q. Zhang,
Competitive exclusion for a two-species chemotaxis system with two chemicals, Appl. Math. Lett., 83 (2018), 27-32.
doi: 10.1016/j.aml.2018.03.012. |
[46] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
[47] |
Q. Zhang, X. Liu and X. Yang, Global existence and asymptotic behavior of solutions to a two-species chemotaxis system with two chemicals, J. Math. Phys., 58 (2017), 111504, 9 pp.
doi: 10.1063/1.5011725. |
[48] |
J. Zhao, C. Mu, D. Zhou and K. Lin,
A parabolic-elliptic-elliptic attraction-repulsion chemotaxis system with logistic source, J. Math. Anal. Appl., 455 (2017), 650-679.
doi: 10.1016/j.jmaa.2017.05.068. |
[49] |
J. Zheng,
Boundedness in a two-species quasi-linear chemotaxis system with two chemicals, Topol. Methods Nonlinear Anal., 49 (2017), 463-480.
doi: 10.12775/TMNA.2016.082. |
[50] |
P. Zheng and C. Mu,
Global boundedness in a two-competing-species chemotaxis system with two chemicals, Acta Appl. Math., 148 (2017), 157-177.
doi: 10.1007/s10440-016-0083-0. |
[51] |
P. Zheng, C. Mu and X. Hu, Persistence property in a two-species chemotaxis system with two signals, J. Math. Phys., 58 (2017), 111501, 17 pp.
doi: 10.1063/1.5010681. |
[52] |
P. Zheng, C. Mu and Y. Mi,
Global stability in a two-competing-species chemotaxis system with two chemicals, Differential Integral Equations, 31 (2018), 547-558.
|
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