# American Institute of Mathematical Sciences

November  2013, 12(6): 3027-3046. doi: 10.3934/cpaa.2013.12.3027

## Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 2 Department of Mathematics, Tulane University, New Orleans, LA 70118

Received  April 2012 Revised  November 2013 Published  May 2013

In the first part of this paper, we investigate the qualitative behavior of classical solutions for a one-dimensional parabolic system derived from a repulsive chemotaxis model on bounded domains. It is shown that classical solutions to the initial-boundary value problem exist globally in time for large data and converge to constant equilibrium states exponentially in time. The results indicate that repulsive chemotaxis exhibits a strong tendency against pattern formation. In the second part, we study diffusion limit and convergence rate of the model toward a non-diffusive problem studied in [11]. It is shown that when the chemical diffusion coefficient $\varepsilon$ tends to zero, the solution is convergent in $L^{\infty}$-norm with respect to $\varepsilon$ at order $O(\varepsilon)$.
Citation: Zhi-An Wang, Kun Zhao. Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model. Communications on Pure & Applied Analysis, 2013, 12 (6) : 3027-3046. doi: 10.3934/cpaa.2013.12.3027
##### References:
 [1] W. Alt and D. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691.  doi: 10.1007/BF00275511.  Google Scholar [2] D. Balding and D. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53.  doi: 10.1016/S0022-5193(85)80255-1.  Google Scholar [3] S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomath., 55 (1984), 61.  doi: 10.1007/978-3-642-45589-6_6.  Google Scholar [4] T. Cieślak, P. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system,, Banach Center Publ., 81 (2008), 105.  doi: 10.4064/bc81-0-7.  Google Scholar [5] J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data,, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629.  doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar [6] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its con- sequences I,, Jahresberichte der DMV, 105 (2003), 103.   Google Scholar [7] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [8] E. Keller and L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis,, J. Theor. Biol., 26 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar [9] H. Levine and B. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 683.  doi: 10.1137/S0036139995291106.  Google Scholar [10] D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci, 21 (2011), 1631.  doi: 10.1142/S0218202511005519.  Google Scholar [11] T. Li, R. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar [12] T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar [13] T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,, Math. Models Methods Appl. Sci., 20 (2010), 1967.   Google Scholar [14] T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310.  doi: 10.1016/j.jde.2010.09.020.  Google Scholar [15] T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model,, Math. Biosci., 240 (2012), 161.  doi: 10.1016/j.mbs.2012.07.003.  Google Scholar [16] C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [17] J. Murray, "Mathematical Biology I: An Introduction,", 3$^{rd}$ edition, (2002).   Google Scholar [18] H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044.  doi: 10.1137/S0036139995288976.  Google Scholar [19] L. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.  doi: 10.1137/0132054.  Google Scholar [20] J. Sherratt, E. Sage and J. Murray, Chemical control of eukaryotic cell movement: a new model,, J. Theor. Biol., 162 (1993), 23.  doi: 10.1006/jtbi.1993.1074.  Google Scholar [21] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Contin. Dyn. Syst - Series B., 18 (2013), 821.   Google Scholar [22] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models. Methods Appli. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar [23] Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.  doi: 10.1002/mma.898.  Google Scholar [24] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar [25] Y. Yang, H. Chen and W. Liu, On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763.  doi: 10.1137/S0036141000337796.  Google Scholar [26] M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2006), 1017.  doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar

show all references

##### References:
 [1] W. Alt and D. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation,, J. Math. Biol., 24 (1987), 691.  doi: 10.1007/BF00275511.  Google Scholar [2] D. Balding and D. McElwain, A mathematical model of tumour-induced capillary growth,, J. Theor. Biol., 114 (1985), 53.  doi: 10.1016/S0022-5193(85)80255-1.  Google Scholar [3] S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomath., 55 (1984), 61.  doi: 10.1007/978-3-642-45589-6_6.  Google Scholar [4] T. Cieślak, P. Laurencot and C. Morales-Rodrigo, Global existence and convergence to steady states in a chemorepulsion system,, Banach Center Publ., 81 (2008), 105.  doi: 10.4064/bc81-0-7.  Google Scholar [5] J. Guo, J. Xiao, H. Zhao and C. Zhu, Global solutions to a hyperbolic-parabolic coupled system with large initial data,, Acta Math. Sci. Ser. B Engl. Ed, 29 (2009), 629.  doi: 10.1016/S0252-9602(09)60059-X.  Google Scholar [6] D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its con- sequences I,, Jahresberichte der DMV, 105 (2003), 103.   Google Scholar [7] E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399.  doi: 10.1016/0022-5193(70)90092-5.  Google Scholar [8] E. Keller and L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis,, J. Theor. Biol., 26 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar [9] H. Levine and B. Sleeman, A system of reaction diffusion equations arising in the theory of reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 683.  doi: 10.1137/S0036139995291106.  Google Scholar [10] D. Li, T. Li and K. Zhao, On a hyperbolic-parabolic system modeling chemotaxis,, Math. Models Methods Appl. Sci, 21 (2011), 1631.  doi: 10.1142/S0218202511005519.  Google Scholar [11] T. Li, R. Pan and K. Zhao, Global dynamics of a chemotaxis model on bounded domains with large data,, SIAM J. Appl. Math., 72 (2012), 417.  doi: 10.1137/110829453.  Google Scholar [12] T. Li and Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis,, SIAM J. Appl. Math., 70 (2009), 1522.  doi: 10.1137/09075161X.  Google Scholar [13] T. Li and Z. Wang, Nonlinear stability of large amplitude viscous shock waves of a hyperbolic-parabolic system arising in chemotaxis,, Math. Models Methods Appl. Sci., 20 (2010), 1967.   Google Scholar [14] T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,, J. Differential Equations, 250 (2011), 1310.  doi: 10.1016/j.jde.2010.09.020.  Google Scholar [15] T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model,, Math. Biosci., 240 (2012), 161.  doi: 10.1016/j.mbs.2012.07.003.  Google Scholar [16] C. Lin, W. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar [17] J. Murray, "Mathematical Biology I: An Introduction,", 3$^{rd}$ edition, (2002).   Google Scholar [18] H. Othmer and A. Stevens, Aggregation, blowup and collapse: The ABC's of taxis in reinforced random walks,, SIAM J. Appl. Math., 57 (1997), 1044.  doi: 10.1137/S0036139995288976.  Google Scholar [19] L. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.  doi: 10.1137/0132054.  Google Scholar [20] J. Sherratt, E. Sage and J. Murray, Chemical control of eukaryotic cell movement: a new model,, J. Theor. Biol., 162 (1993), 23.  doi: 10.1006/jtbi.1993.1074.  Google Scholar [21] Y. Tao, L. Wang and Z. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension,, Discrete Contin. Dyn. Syst - Series B., 18 (2013), 821.   Google Scholar [22] Y. Tao and Z. Wang, Competing effects of attraction vs. repulsion in chemotaxis,, Math. Models. Methods Appli. Sci., 23 (2013), 1.  doi: 10.1142/S0218202512500443.  Google Scholar [23] Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.  doi: 10.1002/mma.898.  Google Scholar [24] M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Math. Methods Appl. Sci., 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar [25] Y. Yang, H. Chen and W. Liu, On existence of global solutions and blow-up to a system of the reaction-diffusion equations modelling chemotaxis,, SIAM J. Math. Anal., 33 (2001), 763.  doi: 10.1137/S0036141000337796.  Google Scholar [26] M. Zhang and C. Zhu, Global existence of solutions to a hyperbolic-parabolic system,, Proc. Amer. Math. Soc., 135 (2006), 1017.  doi: 10.1090/S0002-9939-06-08773-9.  Google Scholar
 [1] Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381 [2] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [3] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 [4] Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020398 [5] Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161 [6] Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354 [7] Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302 [8] Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360 [9] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [10] Yi-Long Luo, Yangjun Ma. Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 921-966. doi: 10.3934/dcds.2020304 [11] Tong Tang, Jianzhu Sun. Local well-posedness for the density-dependent incompressible magneto-micropolar system with vacuum. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020377 [12] Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 [13] Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002 [14] Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 [15] Hui Zhao, Zhengrong Liu, Yiren Chen. Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021011 [16] Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312 [17] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 [18] Xueli Bai, Fang Li. Global dynamics of competition models with nonsymmetric nonlocal dispersals when one diffusion rate is small. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3075-3092. doi: 10.3934/dcds.2020035 [19] Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $\mathbb{R}^n_+$. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033 [20] Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

2019 Impact Factor: 1.105