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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

X. Cao and M. Winkler, Sharp decay estimates in a bioconvection model with quardratic degradation in bounded domains, preprint, 2016. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

X. Cao and M. Winkler, Sharp decay estimates in a bioconvection model with quardratic degradation in bounded domains, preprint, 2016. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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