Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity

Previous Article
On random cocycle attractors with autonomous attraction universes
DCDS-B Home
This Issue
Next Article
Computing stable hierarchies of fiber bundles

November 2017, 22(9): 3369-3378. doi: 10.3934/dcdsb.2017141

Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity

Xinru Cao ^1,2,,

1.	Institute of Mathematical Sciences, Renmin University, Beijing 100872, China
2.	Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany

* Corresponding author

Received October 2016 Revised January 2017 Published April 2017

Fund Project: XC is supported by the Research Funds of Renmin University of China (15XNLF21), and by China Postdoctoral Science Foundation (2016M591319).

The fully parabolic Keller-Segel system with logistic source

$\begin{equation} \left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, &(x,t)\in \Omega\times (0,T),\\ \displaystyle \tau v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T), \end{array} \right.(\star) \end{equation}$

is considered in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N≥ 1$) under Neumann boundary conditions, where $κ∈\mathbb{R}$, $μ>0$, $χ>0$ and $τ>0$. It is shown that if the ratio $\frac{χ}{μ}$ is sufficiently small, then any global classical solution $(u, v)$ converges to the spatially homogenous steady state $(\frac{κ_+}{μ}, \frac{κ_+}{μ})$ in the large time limit. Here we use an approach based on maximal Sobolev regularity and thus remove the restrictions $τ=1$ and the convexity of $\Omega$ required in [17].

Keywords: Chemotaxis, asymptotic behavior, stability.

Mathematics Subject Classification: Primary:35B40, 35K45.

Citation: Xinru Cao. Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3369-3378. doi: 10.3934/dcdsb.2017141

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.

[1]	Georges Chamoun, Moustafa Ibrahim, Mazen Saad, Raafat Talhouk. Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4165-4188. doi: 10.3934/dcdsb.2020092
[2]	Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 465-475. doi: 10.3934/dcdsb.2017021
[3]	Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011
[4]	Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315
[5]	Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6253-6265. doi: 10.3934/dcdsb.2021017
[6]	Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343
[7]	Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437
[8]	Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064
[9]	Lu Xu, Chunlai Mu, Qiao Xin. Global boundedness and asymptotic behavior of solutions for a quasilinear chemotaxis model of multiple sclerosis with nonlinear signal secretion. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022118
[10]	Francesca Romana Guarguaglini, Corrado Mascia, Roberto Natalini, Magali Ribot. Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 39-76. doi: 10.3934/dcdsb.2009.12.39
[11]	Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097
[12]	Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061
[13]	Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 269-278. doi: 10.3934/dcdss.2020015
[14]	Xu Pan, Liangchen Wang. Boundedness and asymptotic stability in a quasilinear two-species chemotaxis system with nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2211-2236. doi: 10.3934/cpaa.2021064
[15]	Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 4077-4095. doi: 10.3934/dcdsb.2021218
[16]	Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721
[17]	Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301
[18]	Rafał Celiński, Andrzej Raczyński. Asymptotic profile of solutions to a certain chemotaxis system. Communications on Pure and Applied Analysis, 2020, 19 (2) : 911-922. doi: 10.3934/cpaa.2020041
[19]	Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure and Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161
[20]	Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Xinru Cao
on Google Scholar
Xinru Cao

2021 Impact Factor: 1.497

Tools

Article outline

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Xinru Cao
on Google Scholar
Xinru Cao

[Back to Top]