American Institute of Mathematical Sciences

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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

Zhi-An Wang 1, and  Kun Zhao 2,

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 $O(\varepsilon)$.
Keywords: classical solution, initial-boundary value problem, diffusion limit, long-time behavior, convergence rate., Chemotaxis, global well-posedness.
Mathematics Subject Classification: Primary: 35K55, 35K57, 35K45, 35K50; Secondary: 92C15, 92C17, 92B9.
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

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