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Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.China |
We are interested in the existence and stability of traveling waves of arbitrary amplitudes to a chemotaxis model with porous medium diffusion. We first make a complete classification of traveling waves under specific relations among the biological parameters. Then we show all these traveling waves are asymptotically stable under appropriate perturbations. The proof is based on a Cole-Hopf transformation and the energy method.
References:
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M. Burger, M. Di Francesco and Y. Dolak-Strub,
The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.
doi: 10.1137/050637923. |
[2] |
S.-H. Choi and Y.-J. Kim, Chemotactic traveling waves with compact support, J. Math. Anal. Appl., 488 (2020), 124090, 21 pp.
doi: 10.1016/j.jmaa.2020.124090. |
[3] |
C. Deng and T. Li,
Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.
doi: 10.1016/j.jde.2014.05.014. |
[4] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[5] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.
|
[6] |
H.-Y. Jin, J. Li and Z.-A. Wang,
Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[7] |
Y. V. Kalinin, L. Jiang, Y. Tu and M. Wu,
Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[8] |
S. Kawashima and A. Matsumura,
Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math., 47 (1994), 1547-1569.
doi: 10.1002/cpa.3160471202. |
[9] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 26 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[10] |
H. A. Levine and B. D. Sleeman,
A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.
doi: 10.1137/S0036139995291106. |
[11] |
D. Li, R. Pan and K. Zhao,
Quantitative decay of a hybrid type chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.
doi: 10.1088/0951-7715/28/7/2181. |
[12] |
J. Li and Z. Wang,
Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space, J. Differential Equations, 268 (2020), 6940-6970.
doi: 10.1016/j.jde.2019.11.076. |
[13] |
T. Li, R. H. Pan and K. Zhao,
Global dynamics of a chemotaxis model on bounded domains with large data, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[14] |
T. Li and Z.-A. Wang,
Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2010), 1522-1541.
doi: 10.1137/09075161X. |
[15] |
V. R. Martinez, Z.-A. Wang and K. Zhao,
Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383-1424.
doi: 10.1512/iumj.2018.67.7394. |
[16] |
A. Matsumura and K. Nishihara,
On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25.
doi: 10.1007/BF03167036. |
[17] |
T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Math'ematiques d'Orsay 78-02, D'epartement de Math'ematique, Universit'e de ParisSud, Orsay, France, 1978. |
[18] |
M. Olson, R. Ford, J. Smith and E. Fernandez,
Quantification of bacterial chemotaxis in porous media using magnetic resonance imaging, Environ. Sci. Technol., 38 (2004), 3864-3870.
doi: 10.1021/es035236s. |
[19] |
H. G. Othmer and A. Stevens,
Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[20] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[21] |
F. Valdés-Parada, M. Porter, K. Narayanaswamy, R. Ford and B. Wood, Upscaling microbial chemotaxis in porous media, Adv. Water Resour., 32 (2009), 1413-1428. Google Scholar |
[22] |
Z.-A. Wang,
Mathematics of traveling waves in chemotaxis: A review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[23] |
Z.-A. Wang and T. Hillen,
Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
[24] |
Z.-A. Wang, Z. Xiang and P. Yu,
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.
doi: 10.1016/j.jde.2015.09.063. |
[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), 763-785.
doi: 10.1137/S0036141000337796. |
[26] |
Y. Yang, H. Chen, W. Liu and B. D. Sleeman,
The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451.
doi: 10.1016/j.jde.2005.01.002. |
[27] |
M. Zhang and C. J. Zhu,
Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.
doi: 10.1090/S0002-9939-06-08773-9. |
show all references
References:
[1] |
M. Burger, M. Di Francesco and Y. Dolak-Strub,
The Keller-Segel model for chemotaxis with prevention of overcrowding: Linear vs. nonlinear diffusion, SIAM J. Math. Anal., 38 (2006), 1288-1315.
doi: 10.1137/050637923. |
[2] |
S.-H. Choi and Y.-J. Kim, Chemotactic traveling waves with compact support, J. Math. Anal. Appl., 488 (2020), 124090, 21 pp.
doi: 10.1016/j.jmaa.2020.124090. |
[3] |
C. Deng and T. Li,
Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.
doi: 10.1016/j.jde.2014.05.014. |
[4] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[5] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69.
|
[6] |
H.-Y. Jin, J. Li and Z.-A. Wang,
Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[7] |
Y. V. Kalinin, L. Jiang, Y. Tu and M. Wu,
Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[8] |
S. Kawashima and A. Matsumura,
Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Comm. Pure Appl. Math., 47 (1994), 1547-1569.
doi: 10.1002/cpa.3160471202. |
[9] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 26 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[10] |
H. A. Levine and B. D. Sleeman,
A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.
doi: 10.1137/S0036139995291106. |
[11] |
D. Li, R. Pan and K. Zhao,
Quantitative decay of a hybrid type chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.
doi: 10.1088/0951-7715/28/7/2181. |
[12] |
J. Li and Z. Wang,
Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space, J. Differential Equations, 268 (2020), 6940-6970.
doi: 10.1016/j.jde.2019.11.076. |
[13] |
T. Li, R. H. Pan and K. Zhao,
Global dynamics of a chemotaxis model on bounded domains with large data, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[14] |
T. Li and Z.-A. Wang,
Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2010), 1522-1541.
doi: 10.1137/09075161X. |
[15] |
V. R. Martinez, Z.-A. Wang and K. Zhao,
Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383-1424.
doi: 10.1512/iumj.2018.67.7394. |
[16] |
A. Matsumura and K. Nishihara,
On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25.
doi: 10.1007/BF03167036. |
[17] |
T. Nishida, Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics, Publications Math'ematiques d'Orsay 78-02, D'epartement de Math'ematique, Universit'e de ParisSud, Orsay, France, 1978. |
[18] |
M. Olson, R. Ford, J. Smith and E. Fernandez,
Quantification of bacterial chemotaxis in porous media using magnetic resonance imaging, Environ. Sci. Technol., 38 (2004), 3864-3870.
doi: 10.1021/es035236s. |
[19] |
H. G. Othmer and A. Stevens,
Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[20] |
Y. Tao and M. Winkler,
Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 1901-1914.
doi: 10.3934/dcds.2012.32.1901. |
[21] |
F. Valdés-Parada, M. Porter, K. Narayanaswamy, R. Ford and B. Wood, Upscaling microbial chemotaxis in porous media, Adv. Water Resour., 32 (2009), 1413-1428. Google Scholar |
[22] |
Z.-A. Wang,
Mathematics of traveling waves in chemotaxis: A review paper, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[23] |
Z.-A. Wang and T. Hillen,
Shock formation in a chemotaxis model, Math. Methods Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
[24] |
Z.-A. Wang, Z. Xiang and P. Yu,
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.
doi: 10.1016/j.jde.2015.09.063. |
[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), 763-785.
doi: 10.1137/S0036141000337796. |
[26] |
Y. Yang, H. Chen, W. Liu and B. D. Sleeman,
The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451.
doi: 10.1016/j.jde.2005.01.002. |
[27] |
M. Zhang and C. J. Zhu,
Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017-1027.
doi: 10.1090/S0002-9939-06-08773-9. |
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