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December  2021, 26(12): 6253-6265. doi: 10.3934/dcdsb.2021017

Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion

Mohammad Ghani , Jingyu Li and  Kaijun Zhang ,

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.China

* Corresponding author: zhangkj201@nenu.edu.cn

Received  August 2020 Revised  November 2020 Published  December 2021 Early access  January 2021

Fund Project: The third author is supported by National Science Foundation of China (No. 11771071)

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.

Keywords: Chemotaxis, nonlinear diffusion, asymptotic stability, traveling fronts.
Mathematics Subject Classification: Primary: 35A01, 35B40, 35Q92; Secondary: 92C17.
Citation: Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6253-6265. doi: 10.3934/dcdsb.2021017
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. 

[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. 

[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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