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Computing stable hierarchies of fiber bundles
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity
1. | Institute of Mathematical Sciences, Renmin University, Beijing 100872, China |
2. | Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany |
$\begin{equation} \left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, &(x,t)\in \Omega\times (0,T),\\ \displaystyle \tau v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T), \end{array} \right.(\star) \end{equation}$ |
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA Journal of Applied Mathematics, 81 (2016), 860-876, arXiv: 1604.03529, 2016.
doi: 10.1093/imamat/hxw036. |
[3] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model Z. Angew. Math. Phys. , 67 (2016), Art. 11, 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[5] |
X. Cao and M. Winkler, Sharp decay estimates in a bioconvection model with quardratic degradation in bounded domains, preprint, 2016. |
[6] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete and Continuous Dynamical Systems-B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[7] |
N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system preprint, 2013.
doi: 10.1016/j.matpur.2013.01.020. |
[8] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[9] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
|
[10] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[11] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[12] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-spcies chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[13] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[14] |
M. Winkler,
Blow-up on a higher-dimensional chemotaxis system deapite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[15] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[16] |
M. Winkler,
How far can chemotatic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[17] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[18] |
C. Yang, X. Cao, Z. Jiang and S. Zheng,
Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.
doi: 10.1016/j.jmaa.2015.04.093. |
show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black, J. Lankeit and M. Mizukami, On the weakly competitive case in a two-species chemotaxis model, IMA Journal of Applied Mathematics, 81 (2016), 860-876, arXiv: 1604.03529, 2016.
doi: 10.1093/imamat/hxw036. |
[3] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[4] |
X. Cao, Boundedness in a three-dimensional chemotaxis-haptotaxis model Z. Angew. Math. Phys. , 67 (2016), Art. 11, 13 pp.
doi: 10.1007/s00033-015-0601-3. |
[5] |
X. Cao and M. Winkler, Sharp decay estimates in a bioconvection model with quardratic degradation in bounded domains, preprint, 2016. |
[6] |
J. Lankeit,
Chemotaxis can prevent thresholds on population density, Discrete and Continuous Dynamical Systems-B, 20 (2015), 1499-1527.
doi: 10.3934/dcdsb.2015.20.1499. |
[7] |
N. Mizoguchi and M. Winkler, Blow-up in the two-dimensional parabolic Keller-Segel system preprint, 2013.
doi: 10.1016/j.matpur.2013.01.020. |
[8] |
J. I. Tello and M. Winkler,
A chemotaxis system with logistic source, Comm. Partial Differential Equations, 32 (2007), 849-877.
doi: 10.1080/03605300701319003. |
[9] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
|
[10] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[11] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[12] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-spcies chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[13] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[14] |
M. Winkler,
Blow-up on a higher-dimensional chemotaxis system deapite logistic growth restriction, J. Math. Anal. Appl., 384 (2011), 261-272.
doi: 10.1016/j.jmaa.2011.05.057. |
[15] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[16] |
M. Winkler,
How far can chemotatic cross-diffusion enforce exceeding carrying capacities?, J. Nonlinear Sci., 24 (2014), 809-855.
doi: 10.1007/s00332-014-9205-x. |
[17] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[18] |
C. Yang, X. Cao, Z. Jiang and S. Zheng,
Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic source, J. Math. Anal. Appl., 430 (2015), 585-591.
doi: 10.1016/j.jmaa.2015.04.093. |
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