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On symmetry results for elliptic equations with convex nonlinearities
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 |
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. |
[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. |
[3] |
S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomath., 55 (1984), 61.
doi: 10.1007/978-3-642-45589-6_6. |
[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. |
[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. |
[6] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its con- sequences I,, Jahresberichte der DMV, 105 (2003), 103.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
J. Murray, "Mathematical Biology I: An Introduction,", 3$^{rd}$ edition, (2002).
|
[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. |
[19] |
L. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.
doi: 10.1137/0132054. |
[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. |
[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. |
[23] |
Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.
doi: 10.1002/mma.898. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] |
S. Childress, Chemotactic collapse in two dimensions,, Lect. Notes in Biomath., 55 (1984), 61.
doi: 10.1007/978-3-642-45589-6_6. |
[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. |
[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. |
[6] |
D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its con- sequences I,, Jahresberichte der DMV, 105 (2003), 103.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
J. Murray, "Mathematical Biology I: An Introduction,", 3$^{rd}$ edition, (2002).
|
[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. |
[19] |
L. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653.
doi: 10.1137/0132054. |
[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. |
[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. |
[23] |
Z. Wang and T. Hillen, Shock formation in a chemotaxis model,, Math. Methods. Appl. Sci., 31 (2008), 45.
doi: 10.1002/mma.898. |
[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. |
[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. |
[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. |
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