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Existence, uniqueness, and stability of bubble solutions of a chemotaxis model
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 |
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. |
[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. |
[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. |
[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. |
[5] |
Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, RIMS Kokyuroku, 1025 (1998), 44-65. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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., ().
|
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. |
[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. |
[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. |
[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. |
[5] |
Y. Kabeya and W.-M. Ni, Stationary Keller-Segel model with the linear sensitivity, RIMS Kokyuroku, 1025 (1998), 44-65. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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., ().
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