# American Institute of Mathematical Sciences

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

## 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

* 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
 $$$\left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, &(x,t)\in \Omega\times (0,T),\\ \displaystyle \tau v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T), \end{array} \right.(\star)$$$
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].
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
 [1] Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 465-475. doi: 10.3934/dcdsb.2017021 [2] Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 203-209. doi: 10.3934/dcdss.2020011 [3] Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437 [4] Telma Silva, Adélia Sequeira, Rafael F. Santos, Jorge Tiago. Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 343-362. doi: 10.3934/dcdss.2016.9.343 [5] 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 & Continuous Dynamical Systems - B, 2009, 12 (1) : 39-76. doi: 10.3934/dcdsb.2009.12.39 [6] Masaaki Mizukami. Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2301-2319. doi: 10.3934/dcdsb.2017097 [7] Tobias Black. Global existence and asymptotic stability in a competitive two-species chemotaxis system with two signals. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1253-1272. doi: 10.3934/dcdsb.2017061 [8] Masaaki Mizukami. Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 269-278. doi: 10.3934/dcdss.2020015 [9] Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721 [10] Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure & Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161 [11] Hunseok Kang. Asymptotic behavior of a discrete turing model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 265-284. doi: 10.3934/dcds.2010.27.265 [12] Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063 [13] Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805 [14] Doan Duy Hai, Atsushi Yagi. Longtime behavior of solutions to chemotaxis-proliferation model with three variables. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3957-3974. doi: 10.3934/dcds.2012.32.3957 [15] Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041 [16] Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009 [17] Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks & Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003 [18] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 [19] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [20] M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

2018 Impact Factor: 1.008