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Dynamic transition and pattern formation for chemotactic systems
1. | Department of Mathematics, Sichuan University, Chengdu |
2. | Department of Mathematics, Indiana University, Bloomington, IN 47405 |
References:
[1] |
M. P. Brenner, L. S. Levitov and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria, Biophysical Journal, 74 (1998), 1677-1693.
doi: 10.1016/S0006-3495(98)77880-4. |
[2] |
E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[3] |
__________, Dynamics of formation of symmetric patterns of chemotactic bacteria, Nature, 376 (1995), 49-53. |
[4] |
Y. Guo and H. J. Hwang, Pattern formation (I): the Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530.
doi: 10.1016/j.jde.2010.07.025. |
[5] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[6] |
H. Liu, T. Sengul and S. Wang, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility, Journal of Mathematical Physics, 53 (2012), 1-31. |
[7] |
T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, 555+XXII pp, 2013.
doi: 10.1007/978-1-4614-8963-4. |
[8] |
_________, Dynamic transition theory for thermohaline circulation, Physica D, 239 (2010), 167-189. |
[9] |
P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), vi+71.
doi: 10.1090/S0065-9266-09-00568-7. |
[10] |
J. Murray, Mathematical Biology, II, 3rd Ed. Springer-Verlag, 2002. |
[11] |
G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound., 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[12] |
B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.
doi: 10.1090/S0002-9947-08-04656-4. |
[13] |
B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: Existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270.
doi: 10.1088/0951-7715/24/4/012. |
show all references
References:
[1] |
M. P. Brenner, L. S. Levitov and E. O. Budrene, Physical mechanisms for chemotactic pattern formation by bacteria, Biophysical Journal, 74 (1998), 1677-1693.
doi: 10.1016/S0006-3495(98)77880-4. |
[2] |
E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of Escherichia coli, Nature, 349 (1991), 630-633.
doi: 10.1038/349630a0. |
[3] |
__________, Dynamics of formation of symmetric patterns of chemotactic bacteria, Nature, 376 (1995), 49-53. |
[4] |
Y. Guo and H. J. Hwang, Pattern formation (I): the Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530.
doi: 10.1016/j.jde.2010.07.025. |
[5] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[6] |
H. Liu, T. Sengul and S. Wang, Dynamic transitions for quasilinear systems and Cahn-Hilliard equation with Onsager mobility, Journal of Mathematical Physics, 53 (2012), 1-31. |
[7] |
T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, 555+XXII pp, 2013.
doi: 10.1007/978-1-4614-8963-4. |
[8] |
_________, Dynamic transition theory for thermohaline circulation, Physica D, 239 (2010), 167-189. |
[9] |
P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), vi+71.
doi: 10.1090/S0065-9266-09-00568-7. |
[10] |
J. Murray, Mathematical Biology, II, 3rd Ed. Springer-Verlag, 2002. |
[11] |
G. Nadin, B. Perthame and L. Ryzhik, Traveling waves for the Keller-Segel system with Fisher birth terms, Interfaces Free Bound., 10 (2008), 517-538.
doi: 10.4171/IFB/200. |
[12] |
B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335.
doi: 10.1090/S0002-9947-08-04656-4. |
[13] |
B. Perthame, C. Schmeiser, M. Tang and N. Vauchelet, Travelling plateaus for a hyperbolic Keller-Segel system with attraction and repulsion: Existence and branching instabilities, Nonlinearity, 24 (2011), 1253-1270.
doi: 10.1088/0951-7715/24/4/012. |
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