American Institute of Mathematical Sciences

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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 - A, 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.  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.  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.   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.   Google Scholar

[5]

Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity,, RIMS Kokyuroku, 1025 (1998), 44.   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.  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.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

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[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.  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.  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.  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.  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.  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.   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.   Google Scholar

[5]

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

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