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Stability of singular limit cycles for Abel equations
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces
1. | School of Mathematical Science, Dalian University of Technology, Dalian 116023, China |
Our approach allows us to furthermore study a general chemotaxis system with rotational sensitivity in dimension 2, which is lacking the natural energy structure associated with ($\star$). For such systems, we prove a global existence and boundedness result under corresponding smallness conditions on the initially present total mass of cells and the chemical gradient.
References:
[1] |
X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation,, Nonlinearity, 27 (2014), 1899.
doi: 10.1088/0951-7715/27/8/1899. |
[2] |
L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math, 72 (2004), 1.
doi: 10.1007/s00032-003-0026-x. |
[3] |
L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47 (2008), 755.
doi: 10.1016/j.mcm.2007.06.005. |
[4] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24 (1997), 633.
|
[5] |
T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis,, J. Math. Biology, 58 (2009), 183.
doi: 10.1007/s00285-008-0201-3. |
[6] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jber. DMV, 105 (2003), 103.
|
[7] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.
doi: 10.1016/j.jde.2004.10.022. |
[8] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Tran. Amer. Math. Soc., 329 (1992), 819.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[9] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.
doi: 10.1016/0022-5193(70)90092-5. |
[10] |
O. A. Ladyžxenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, ().
|
[11] |
T. Li, A. Suen, C. Xue and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms,, Math. Models Methods Appl. Sci., (). Google Scholar |
[12] |
T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains,, J. Inequal. Appl., 6 (2001), 37.
doi: 10.1155/S1025583401000042. |
[13] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.
|
[14] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl. 2, 2 (1997).
doi: 10.1115/1.3424338. |
[15] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines,, Proc. Nat. Acad. Sci, 102 (2005), 2277.
doi: 10.1073/pnas.0406724102. |
[16] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.
doi: 10.1016/j.jde.2010.02.008. |
[17] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.
doi: 10.1016/j.matpur.2013.01.020. |
[18] |
C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133.
doi: 10.1137/070711505. |
show all references
References:
[1] |
X. Cao and S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation,, Nonlinearity, 27 (2014), 1899.
doi: 10.1088/0951-7715/27/8/1899. |
[2] |
L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions,, Milan J. Math, 72 (2004), 1.
doi: 10.1007/s00032-003-0026-x. |
[3] |
L. Corrias and B. Perthame, Asymptotic decay for the solutions of the parabolic-parabolic Keller-Segel chemotaxis system in critical spaces,, Mathematical and Computer Modelling, 47 (2008), 755.
doi: 10.1016/j.mcm.2007.06.005. |
[4] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore, 24 (1997), 633.
|
[5] |
T. Hillen and K. J. Painter, A user's guide to PDE models in a chemotaxis,, J. Math. Biology, 58 (2009), 183.
doi: 10.1007/s00285-008-0201-3. |
[6] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I,, Jber. DMV, 105 (2003), 103.
|
[7] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52.
doi: 10.1016/j.jde.2004.10.022. |
[8] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Tran. Amer. Math. Soc., 329 (1992), 819.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[9] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.
doi: 10.1016/0022-5193(70)90092-5. |
[10] |
O. A. Ladyžxenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society, ().
|
[11] |
T. Li, A. Suen, C. Xue and M. Winkler, Global small-data solutions of a two-dimensional chemotaxis system with rotational flux terms,, Math. Models Methods Appl. Sci., (). Google Scholar |
[12] |
T. Nagai, Blowup of Nonradial Solutions to Parabolic-Elliptic Systems Modeling Chemotaxis in Two-Dimensional Domains,, J. Inequal. Appl., 6 (2001), 37.
doi: 10.1155/S1025583401000042. |
[13] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.
|
[14] |
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis,, Stud. Math. Appl. 2, 2 (1997).
doi: 10.1115/1.3424338. |
[15] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines,, Proc. Nat. Acad. Sci, 102 (2005), 2277.
doi: 10.1073/pnas.0406724102. |
[16] |
M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, J. Differential Equations, 248 (2010), 2889.
doi: 10.1016/j.jde.2010.02.008. |
[17] |
M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system,, J. Math. Pures Appl., 100 (2013), 748.
doi: 10.1016/j.matpur.2013.01.020. |
[18] |
C. Xue and H. G. Othmer, Multiscale models of taxis-driven patterning in bacterial population,, SIAM J. Appl. Math., 70 (2009), 133.
doi: 10.1137/070711505. |
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