American Institute of Mathematical Sciences

Advanced
February  2016, 36(2): 805-832. doi: 10.3934/dcds.2016.36.805

Existence, uniqueness, and stability of bubble solutions of a chemotaxis model

Xin Lai 1, , Xinfu Chen 2, , Mingxin Wang 3, , Cong Qin 4, and  Yajing Zhang 5,

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China
2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260
3. 

Natural Science Research Center, Harbin Institute of Technology, Harbin 150080
4. 

Center for Financial Engineering, Soochow University, Suzhou, 215006, China
5. 

School of Mathematical Sciences, Shanxi University, Taiyuan, 030006

Received  June 2014 Revised  July 2014 Published  August 2015

Existence, uniqueness, and stability of Heaviside function like solutions of a Keller and Segel's minimal chemotaxis model are established when a chemotaxis parameter is large enough. Asymptotic expansions of the solution in terms of the large chemotaxis parameter are also derived.
Keywords: chemotaxis., asymptotic expansion, stability, uniqueness, Existence, eigenvalue.
Mathematics Subject Classification: 92C17, 35B30, 35B35, 35G50, 35P2.
Citation: Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805
References:
[1]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of the Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134. doi: 10.1016/j.jde.2014.06.008.  Google Scholar

[2]

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.  Google Scholar

[3]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequence I, Jahresbericht der Deutsche Mathematiker-Vereinigung, 105 (2003), 103-165.  Google Scholar

[4]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequence II, Jahresbericht der Deutsche Mathematiker-Vereinigung, 106 (2004), 51-69.  Google Scholar

[5]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, RIMS Kokyuroku, 1025 (1998), 44-65.  Google Scholar

[6]

K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the 1D Keller-Segel model, IMA Journal of Applied Mathematics, 72 (2007), 140-162. doi: 10.1093/imamat/hxl028.  Google Scholar

[7]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[8]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[9]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Mathematical Journal, 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[10]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, Journal of Mathematical Biology, 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.  Google Scholar

[11]

Y. Zhang, X. Chen, J. Hao, X. Lai and C. Qin, An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model, Journal of Mathematical Analysis and Applications, 420 (2014), 684-704. doi: 10.1016/j.jmaa.2014.06.005.  Google Scholar

[12]

Y. Zhang, X. Chen, J. Hao, X. Lai and C. Qin, An eigenvaue problem arising from stability of a bubble steady state of the Keller-Segel's minimal chemotaxis model,, in preparation., ().   Google Scholar

show all references

References:
[1]

X. Chen, J. Hao, X. Wang, Y. Wu and Y. Zhang, Stability of spiky solution of the Keller-Segel's minimal chemotaxis model, Journal of Differential Equations, 257 (2014), 3102-3134. doi: 10.1016/j.jde.2014.06.008.  Google Scholar

[2]

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.  Google Scholar

[3]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequence I, Jahresbericht der Deutsche Mathematiker-Vereinigung, 105 (2003), 103-165.  Google Scholar

[4]

D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequence II, Jahresbericht der Deutsche Mathematiker-Vereinigung, 106 (2004), 51-69.  Google Scholar

[5]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, RIMS Kokyuroku, 1025 (1998), 44-65.  Google Scholar

[6]

K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in the 1D Keller-Segel model, IMA Journal of Applied Mathematics, 72 (2007), 140-162. doi: 10.1093/imamat/hxl028.  Google Scholar

[7]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[8]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, Journal of Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[9]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Mathematical Journal, 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[10]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly compactness theorem, Journal of Mathematical Biology, 66 (2013), 1241-1266. doi: 10.1007/s00285-012-0533-x.  Google Scholar

[11]

Y. Zhang, X. Chen, J. Hao, X. Lai and C. Qin, An eigenvalue problem arising from spiky steady states of a minimal chemotaxis model, Journal of Mathematical Analysis and Applications, 420 (2014), 684-704. doi: 10.1016/j.jmaa.2014.06.005.  Google Scholar

[12]

Y. Zhang, X. Chen, J. Hao, X. Lai and C. Qin, An eigenvaue problem arising from stability of a bubble steady state of the Keller-Segel's minimal chemotaxis model,, in preparation., ().   Google Scholar

[1]

Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[2]

Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
[3]

Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057
[4]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
[5]

Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[6]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017
[7]

Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725
[8]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220
[9]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793
[10]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437
[11]

Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064
[12]

Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[13]

Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015
[14]

Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064
[15]

Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021218
[16]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
[17]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[18]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[19]

Tómas Chacón-Rebollo, Macarena Gómez-Mármol, Samuele Rubino. On the existence and asymptotic stability of solutions for unsteady mixing-layer models. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 421-436. doi: 10.3934/dcds.2014.34.421
[20]

Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (88)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences