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Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China, China |
References:
[1] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.
|
[2] |
X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces,, Discrete Contin. Dyn. Syst., 35 (2015), 1891.
doi: 10.3934/dcds.2015.35.1891. |
[3] |
T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.
doi: 10.1016/j.jde.2012.01.045. |
[4] |
T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., 129 (2014), 135.
doi: 10.1007/s10440-013-9832-5. |
[5] |
T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models,, J. Differential Equations, 258 (2015), 2080.
doi: 10.1016/j.jde.2014.12.004. |
[6] |
H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.
doi: 10.1002/mana.19981950106. |
[7] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.
|
[8] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159.
doi: 10.1017/S0956792501004363. |
[9] |
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. |
[10] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[11] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.
|
[13] |
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. |
[14] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.
|
[15] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441.
|
[16] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.
doi: 10.1016/j.jde.2011.08.019. |
[17] |
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. |
[18] |
M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12.
doi: 10.1002/mma.1146. |
[19] |
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. |
show all references
References:
[1] |
P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis,, Adv. Math. Sci. Appl., 8 (1998), 715.
|
[2] |
X. Cao, Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces,, Discrete Contin. Dyn. Syst., 35 (2015), 1891.
doi: 10.3934/dcds.2015.35.1891. |
[3] |
T. Cieślak and C. Stinner, Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions,, J. Differential Equations, 252 (2012), 5832.
doi: 10.1016/j.jde.2012.01.045. |
[4] |
T. Cieślak and C. Stinner, Finite-time blowup in a supercritical quasilinear parabolic-parabolic Keller-Segel system in dimension 2,, Acta Appl. Math., 129 (2014), 135.
doi: 10.1007/s10440-013-9832-5. |
[5] |
T. Cieślak and C. Stinner, New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models,, J. Differential Equations, 258 (2015), 2080.
doi: 10.1016/j.jde.2014.12.004. |
[6] |
H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.
doi: 10.1002/mana.19981950106. |
[7] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633.
|
[8] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions,, European J. Appl. Math., 12 (2001), 159.
doi: 10.1017/S0956792501004363. |
[9] |
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. |
[10] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis,, Trans. Amer. Math. Soc., 329 (1992), 819.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[11] |
E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theoret. Biol., 26 (1970), 399.
doi: 10.1016/0022-5193(70)90092-5. |
[12] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system,, Adv. Math. Sci. Appl., 5 (1995), 581.
|
[13] |
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. |
[14] |
T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.
|
[15] |
K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial. Ekvac., 44 (2001), 441.
|
[16] |
Y. Tao and M. Winkler, Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity,, J. Differential Equations, 252 (2012), 692.
doi: 10.1016/j.jde.2011.08.019. |
[17] |
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. |
[18] |
M. Winkler, Does a 'volume-filling effect' always prevent chemotactic collapse?,, Math. Methods Appl. Sci., 33 (2010), 12.
doi: 10.1002/mma.1146. |
[19] |
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. |
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