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The stability of bifurcating steady states of several classes of chemotaxis systems
1. | Department of Basic Courses, Beijing Union University, Beijing 100101 |
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] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
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M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch.Rational Mech.Anal, 52 (1973), 161-180. |
[5] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[6] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165. |
[7] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69. |
[8] |
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
X. Lai, X. Chen, C. Qin and Y. Zhang, Existence, uniqueness, and stability of bubble solutions of a chemotaxis model, preprint. |
[10] |
A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. of Dynamics and Diff. Eqs., 17 (2005), 293-330.
doi: 10.1007/s10884-005-2938-3. |
[11] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[12] |
B. Sleeman, M. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[13] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
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] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[3] |
M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[4] |
M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch.Rational Mech.Anal, 52 (1973), 161-180. |
[5] |
T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[6] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165. |
[7] |
D. Horstmann, From 1970 until now: The Keller-Segal model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69. |
[8] |
E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
X. Lai, X. Chen, C. Qin and Y. Zhang, Existence, uniqueness, and stability of bubble solutions of a chemotaxis model, preprint. |
[10] |
A. B. Potapov and T. Hillen, Metastability in chemotaxis models, J. of Dynamics and Diff. Eqs., 17 (2005), 293-330.
doi: 10.1007/s10884-005-2938-3. |
[11] |
R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc., 292 (1985), 531-556.
doi: 10.1090/S0002-9947-1985-0808736-1. |
[12] |
B. Sleeman, M. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[13] |
X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem, J. Math. Biol., 66 (2013), 1241-1266.
doi: 10.1007/s00285-012-0533-x. |
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