Analysis of traveling waves for nonlinear degenerate viscosity of chemotaxis model under general perturbations

Mohammad Ghani; Mohammad Ghani

doi:10.3934/math.2024068

AIMS Mathematics

2024, Volume 9, Issue 1: 1373-1402. doi: 10.3934/math.2024068

Previous Article Next Article

Research article

Analysis of traveling waves for nonlinear degenerate viscosity of chemotaxis model under general perturbations

Mohammad Ghani ^,

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

Received: 07 November 2023 Revised: 05 December 2023 Accepted: 07 December 2023 Published: 08 December 2023
MSC : 35A01, 35B40, 35Q92, 92C17

In this paper, we generalized the results of the following chemotaxis model with the nonlinear degenerate viscosity

$ \begin{equation*} \begin{cases} u_{t} -\chi (uv)_{x} = D(u^{m})_{xx}, \\ v_{t} -u_{x} = 0, \end{cases} \end{equation*} $

by introducing the following general initial perturbation

$ \begin{equation*} \begin{split} \int_{-\infty}^{+\infty}\kappa(Z_0|\tilde{Z})dx<\infty, \end{split} \end{equation*} $

where $ \kappa $ is the relative entropy function defined in Eq (2.24). We further employed the relative entropy method by choosing the specific shift function. According to the estimates with the cutoff version, and overcoming the complexity caused by the porous media diffusion, the nonlinear orbital stability of traveling waves was established under small amplitude and general perturbations.
- chemotaxis,
- nonlinear diffusion,
- orbital stability,
- energy estimates cutoff,
- general perturbations
Citation: Mohammad Ghani. Analysis of traveling waves for nonlinear degenerate viscosity of chemotaxis model under general perturbations[J]. AIMS Mathematics, 2024, 9(1): 1373-1402. doi: 10.3934/math.2024068

Related Papers:

Abstract

In this paper, we generalized the results of the following chemotaxis model with the nonlinear degenerate viscosity

$ \begin{equation*} \begin{cases} u_{t} -\chi (uv)_{x} = D(u^{m})_{xx}, \\ v_{t} -u_{x} = 0, \end{cases} \end{equation*} $

by introducing the following general initial perturbation

$ \begin{equation*} \begin{split} \int_{-\infty}^{+\infty}\kappa(Z_0|\tilde{Z})dx<\infty, \end{split} \end{equation*} $

where $ \kappa $ is the relative entropy function defined in Eq (2.24). We further employed the relative entropy method by choosing the specific shift function. According to the estimates with the cutoff version, and overcoming the complexity caused by the porous media diffusion, the nonlinear orbital stability of traveling waves was established under small amplitude and general perturbations.

References

[1]	M. Burger, M. Di Francesco, 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. http://dx.doi.org/10.1137/050637923 doi: 10.1137/050637923
[2]	K. Choi, M. Kang, Y. Kwon, A. Vasseur, Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model, Math. Mod. Meth. Appl. S., 30 (2020), 387–437. http://dx.doi.org/10.1142/S0218202520500104 doi: 10.1142/S0218202520500104
[3]	K. Choi, M. Kang, A. Vasseur, Global well-posedness of large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model, J. Math. Pure. Appl., 142 (2020), 266–297. http://dx.doi.org/10.1016/j.matpur.2020.03.002 doi: 10.1016/j.matpur.2020.03.002
[4]	S. Choi, Y. Kim, Chemotactic traveling waves with compact support, J. Math. Anal. Appl., 488 (2020), 124090. http://dx.doi.org/10.1016/j.jmaa.2020.124090 doi: 10.1016/j.jmaa.2020.124090
[5]	C. Deng, T. Li, Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the Sobolev space framework, J. Differ. Equations, 257 (2014), 1311–1332. http://dx.doi.org/10.1016/j.jde.2014.05.014 doi: 10.1016/j.jde.2014.05.014
[6]	M. Ghani, J. Li, K. Zhang, Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion, Discrete Cont. Dyn.-B, 26 (2021), 6253–6265. http://dx.doi.org/10.3934/dcdsb.2021017 doi: 10.3934/dcdsb.2021017
[7]	J. Goodman, A. Szepessy, K. Zumbrun, A Remarks on the stability of viscous shock waves, SIAM J. Math. Anal., 25 (1994), 1463–1467. http://dx.doi.org/10.1137/S0036141092239648 doi: 10.1137/S0036141092239648
[8]	T. Hillen, K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26 (2001), 280–301. http://dx.doi.org/10.1006/aama.2001.0721 doi: 10.1006/aama.2001.0721
[9]	D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences: Ⅰ, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103–165.
[10]	H. Jin, J. Li, Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differ. Equations, 255 (2013), 193–219. http://dx.doi.org/10.1016/j.jde.2013.04.002 doi: 10.1016/j.jde.2013.04.002
[11]	Y. Kalinin, L. Jiang, Y. Tu, M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophys. J., 96 (2009), 2439–2448. http://dx.doi.org/10.1016/j.bpj.2008.10.027 doi: 10.1016/j.bpj.2008.10.027
[12]	S. Kawashima, A. Matsumura, Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pur. Appl. Math., 47 (1994), 1547–1569. http://dx.doi.org/10.1002/cpa.3160471202 doi: 10.1002/cpa.3160471202
[13]	E. Keller, L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theor. Biol., 30 (1971), 235–248. http://dx.doi.org/10.1016/0022-5193(71)90051-8 doi: 10.1016/0022-5193(71)90051-8
[14]	D. Li, R. Pan, K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 28 (2015), 2181. http://dx.doi.org/10.1088/0951-7715/28/7/2181 doi: 10.1088/0951-7715/28/7/2181
[15]	J. Li, Z. Wang, Convergence to traveling waves of a singular PDE-ODE hybrid chemotaxis system in the half space, J. Differ. Equations, 268 (2020), 6940–6970. http://dx.doi.org/10.1016/j.jde.2019.11.076 doi: 10.1016/j.jde.2019.11.076
[16]	T. Li, R. H. Pan, K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417–443. http://dx.doi.org/10.1137/110829453 doi: 10.1137/110829453
[17]	T. Li, Z. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522–1541. http://dx.doi.org/10.1137/09075161X doi: 10.1137/09075161X
[18]	V. Martinez, Z. Wang, K. Zhao, Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana U. Math. J., 67 (2018), 1383–1424.
[19]	M. Olson, R. Ford, J. Smith, E. Fernandez, Quantification of bacterial chemotaxis in porous media using magnetic resonance imaging, Environ. Sci. Technol., 38 (2004), 3864–3870. http://dx.doi.org/10.1021/es035236s doi: 10.1021/es035236s
[20]	B. Sleeman, H. Levine, A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683–730. http://dx.doi.org/10.1137/S0036139995291106 doi: 10.1137/S0036139995291106
[21]	A. Stevens, H. Othmer, Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044–1081. http://dx.doi.org/10.1137/S0036139995288976 doi: 10.1137/S0036139995288976
[22]	Y. Tao, M. Winkler, Global existence and boundedness in a Keller-Segel-Stokes model with any porous medium diffusion, Discrete Cont. Dyn.-A, 32 (2012), 1901–1914. http://dx.doi.org/10.3934/dcds.2012.32.1901 doi: 10.3934/dcds.2012.32.1901
[23]	F. Valdaes-Parada, M. Porter, K. Narayanaswamy, R. Ford, B. Wood, Upscaling microbial chemotaxis in porous media, Adv. Water Resour., 32 (2009), 1413–1428. http://dx.doi.org/10.1016/j.advwatres.2009.06.010 doi: 10.1016/j.advwatres.2009.06.010
[24]	Z. Wang, Mathematics of traveling waves in chemotaxis-review paper, Discrete Cont. Dyn.-B, 18 (2013), 601–641. http://dx.doi.org/10.3934/dcdsb.2013.18.601 doi: 10.3934/dcdsb.2013.18.601
[25]	Z. Wang, Z. Xiang, P. Yu, Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differ. Equations, 260 (2016), 2225–2258. http://dx.doi.org/10.1016/j.jde.2015.09.063 doi: 10.1016/j.jde.2015.09.063
[26]	Z. Wang, T. Hillen, Shock formation in a chemotaxis model, Math. Method. Appl. Sci., 31 (2008), 45–70. http://dx.doi.org/10.1002/mma.898 doi: 10.1002/mma.898
[27]	Y. Yang, H. Chen, 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. http://dx.doi.org/10.1137/S0036141000337796 doi: 10.1137/S0036141000337796
[28]	Y. Yang, H. Chen, W. Liu, B. Sleeman, The solvability of some chemotaxis systems, J. Differ. Equations, 212 (2005), 432–451. http://dx.doi.org/10.1016/j.jde.2005.01.002 doi: 10.1016/j.jde.2005.01.002
[29]	M. Zhang, C. Zhu, Global existence of solutions to a hyperbolic-parabolic system, Proc. Amer. Math. Soc., 135 (2007), 1017–1027. http://dx.doi.org/10.1090/S0002-9939-06-08773-9 doi: 10.1090/S0002-9939-06-08773-9

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)