The stability of bifurcating steady states of several classes of chemotaxis systems

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January 2015, 20(1): 231-248. doi: 10.3934/dcdsb.2015.20.231

The stability of bifurcating steady states of several classes of chemotaxis systems

1.	Department of Basic Courses, Beijing Union University, Beijing 100101

Received October 2013 Revised July 2014 Published November 2014

Abstract
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This paper concerns with the stability of bifurcating steady states obtained in [13] of several chemotaxis systems. By spectral analysis and the principle of the linearized stability, we prove that the bifurcating steady states are stable when the parameters satisfy some certain conditions.

Keywords: Stability, spectral analysis, bifurcating steady states, chemotaxis systems., expansion.

Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 92C1.

Citation: Qian Xu. The stability of bifurcating steady states of several classes of chemotaxis systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 231-248. doi: 10.3934/dcdsb.2015.20.231

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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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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