doi: 10.3934/dcdsb.2022146
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Asymptotic behaviors and existence of traveling wave solutions to the Keller-Segel model with logarithmic sensitivity

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

*Corresponding author: Zengji Du

Received  April 2022 Revised  June 2022 Early access July 2022

In this paper, we are concerned with the existence of traveling wave solutions for the Keller-Segel model with logarithmic sensitivity. By the Hopf-Cole transformation and traveling wave transformation, the degenerate Keller-Segel system is transformed into a singularly perturbed system. By constructing an invariant region to prove the existence of the traveling wave solutions for the degenerate system, we obtain the traveling wave solutions for Keller-Segel system with small parameter by using geometric singular perturbation theory and Fredholm theory. Finally, we discuss the asymptotic behaviors of the traveling wave solutions.

Citation: Chen Li, Jiang Liu, Zengji Du. Asymptotic behaviors and existence of traveling wave solutions to the Keller-Segel model with logarithmic sensitivity. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022146
References:
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J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. 

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S. AiW. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst., 20 (2015), 1-21. 

[3]

J. CarrilloJ. Li and Z. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability, Proc. Lond. Math. Soc., 122 (2021), 42-68.  doi: 10.1112/plms.12319.

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C. Chang, Y. Chen, J. Hong, et al, Existence and instability of traveling pulses of Keller-Segel system with nonlinear chemical gradients and small diffusions, Nonlinearity, 32 (2019), 143–167. doi: 10.1088/1361-6544/aae731.

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C. Deng and T. Li, Global existence and large time behavior of a 2D Keller-Segel system in logarithmic lebesgue spaces, Discrete Contin. Dyn. Syst., 24 (2019), 183-195. 

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Z. DuJ. Liu and Y. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differ. Equ., 270 (2021), 1019-1042.  doi: 10.1016/j.jde.2020.09.009.

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N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

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M. Freitag, Global existence and boundedness in a chemorepulsion system with superlinear diffusion, Discrete Contin. Dyn. Syst., 38 (2018), 5943-5961.  doi: 10.3934/dcds.2018258.

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Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl., 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008.

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Q. HouZ. Wang and K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differ. Equ., 261 (2016), 5035-5070.  doi: 10.1016/j.jde.2016.07.018.

[12]

H. JinJ. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differ. Equ., 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.

[13]

H. Jin and Z. Wang, The Keller-Segel system with logistic growth and signal-dependent motility, Discrete Contin. Dyn. Syst. Ser. B., 26 (2021), 3023-3041.  doi: 10.3934/dcdsb.2020218.

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C. K. R. T. Jones, Geometrical Singular Perturbation Theory, Springer, Berlin, 1995. doi: 10.1007/BFb0095239.

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E. Keller and G. Odell, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243-247. 

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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. 

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J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[18]

T. Li and J. Park, Traveling waves in a chemotaxis model with logistic growth, Discrete Contin. Dyn. Syst. Ser. B., 12 (2019), 6465-6480. 

[19]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differ. Equ., 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.

[20]

T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161-168.  doi: 10.1016/j.mbs.2012.07.003.

[21]

V. MartinezZ. 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.

[22]

H. Peng and Z. Wang, On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity, J. Differ. Equ., 268 (2020), 4374-4415.  doi: 10.1016/j.jde.2019.10.025.

[23]

H. Peng, Z. Wang, K. Zhao, et al, Boundary layers and stabilization of the singular Keller-Segel system, Kinet. Relat. Models, 11 (2018), 1085–1123. doi: 10.3934/krm.2018042.

[24]

J. Rugamba and Y. Zeng, Green's function of the linearized logarithmic Keller-Segel-Fisher/KPP system, Math. Comput. Appl., 23 (2018), Paper No. 56, 12 pp.

[25]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.

[26]

B. Tobias, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal., 50 (2018), 4087-4116.  doi: 10.1137/17M1159488.

[27]

M. Winkler, Unlimited growth in logarithmic Keller-Segel systems, J. Differ. Equ., 309 (2022), 74-97.  doi: 10.1016/j.jde.2021.11.026.

[28]

M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differ. Equ., 264 (2018), 2310-2350.  doi: 10.1016/j.jde.2017.10.029.

[29]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differ. Equ., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[30]

Y. Zeng, Nonlinear stability of diffusive contact wave for a chemotaxis model, J. Differ. Equ., 308 (2022), 286-326.  doi: 10.1016/j.jde.2021.11.008.

[31]

Y. Zeng and K. Zhao, On the logarithmic Keller-Segel-Fisher/KPP system, Discrete Contin. Dyn. Syst., 39 (2019), 5365-5402.  doi: 10.3934/dcds.2019220.

[32]

Y. Zeng and K. Zhao, Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate, J. Differ. Equ., 268 (2020), 1379-1411. 

[33]

X. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source, J. Differ. Equ., 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.

show all references

References:
[1]

J. Adler, Chemoreceptors in bacteria, Science, 166 (1969), 1588-1597. 

[2]

S. AiW. Huang and Z. Wang, Reaction, diffusion and chemotaxis in wave propagation, Discrete Contin. Dyn. Syst., 20 (2015), 1-21. 

[3]

J. CarrilloJ. Li and Z. Wang, Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability, Proc. Lond. Math. Soc., 122 (2021), 42-68.  doi: 10.1112/plms.12319.

[4]

C. Chang, Y. Chen, J. Hong, et al, Existence and instability of traveling pulses of Keller-Segel system with nonlinear chemical gradients and small diffusions, Nonlinearity, 32 (2019), 143–167. doi: 10.1088/1361-6544/aae731.

[5]

C. Conley and R. Gardner, An application of the generalized morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math.J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018.

[6]

C. Deng and T. Li, Global existence and large time behavior of a 2D Keller-Segel system in logarithmic lebesgue spaces, Discrete Contin. Dyn. Syst., 24 (2019), 183-195. 

[7]

Z. DuJ. Liu and Y. Ren, Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach, J. Differ. Equ., 270 (2021), 1019-1042.  doi: 10.1016/j.jde.2020.09.009.

[8]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[9]

M. Freitag, Global existence and boundedness in a chemorepulsion system with superlinear diffusion, Discrete Contin. Dyn. Syst., 38 (2018), 5943-5961.  doi: 10.3934/dcds.2018258.

[10]

Q. Hou and Z. Wang, Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures Appl., 130 (2019), 251-287.  doi: 10.1016/j.matpur.2019.01.008.

[11]

Q. HouZ. Wang and K. Zhao, Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differ. Equ., 261 (2016), 5035-5070.  doi: 10.1016/j.jde.2016.07.018.

[12]

H. JinJ. Li and Z. Wang, Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differ. Equ., 255 (2013), 193-219.  doi: 10.1016/j.jde.2013.04.002.

[13]

H. Jin and Z. Wang, The Keller-Segel system with logistic growth and signal-dependent motility, Discrete Contin. Dyn. Syst. Ser. B., 26 (2021), 3023-3041.  doi: 10.3934/dcdsb.2020218.

[14]

C. K. R. T. Jones, Geometrical Singular Perturbation Theory, Springer, Berlin, 1995. doi: 10.1007/BFb0095239.

[15]

E. Keller and G. Odell, Traveling bands of chemotactic bacteria revisited, J. Theor. Biol., 56 (1976), 243-247. 

[16]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. 

[17]

J. Lankeit, Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic source, J. Differ. Equ., 258 (2015), 1158-1191.  doi: 10.1016/j.jde.2014.10.016.

[18]

T. Li and J. Park, Traveling waves in a chemotaxis model with logistic growth, Discrete Contin. Dyn. Syst. Ser. B., 12 (2019), 6465-6480. 

[19]

T. Li and Z. Wang, Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differ. Equ., 250 (2011), 1310-1333.  doi: 10.1016/j.jde.2010.09.020.

[20]

T. Li and Z. Wang, Steadily propagating waves of a chemotaxis model, Math. Biosci., 240 (2012), 161-168.  doi: 10.1016/j.mbs.2012.07.003.

[21]

V. MartinezZ. 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.

[22]

H. Peng and Z. Wang, On a parabolic-hyperbolic chemotaxis system with discontinuous data: Well-posedness, stability and regularity, J. Differ. Equ., 268 (2020), 4374-4415.  doi: 10.1016/j.jde.2019.10.025.

[23]

H. Peng, Z. Wang, K. Zhao, et al, Boundary layers and stabilization of the singular Keller-Segel system, Kinet. Relat. Models, 11 (2018), 1085–1123. doi: 10.3934/krm.2018042.

[24]

J. Rugamba and Y. Zeng, Green's function of the linearized logarithmic Keller-Segel-Fisher/KPP system, Math. Comput. Appl., 23 (2018), Paper No. 56, 12 pp.

[25]

Y. Tao and M. Winkler, Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.  doi: 10.1007/s00033-015-0541-y.

[26]

B. Tobias, Global very weak solutions to a chemotaxis-fluid system with nonlinear diffusion, SIAM J. Math. Anal., 50 (2018), 4087-4116.  doi: 10.1137/17M1159488.

[27]

M. Winkler, Unlimited growth in logarithmic Keller-Segel systems, J. Differ. Equ., 309 (2022), 74-97.  doi: 10.1016/j.jde.2021.11.026.

[28]

M. Winkler, Renormalized radial large-data solutions to the higher-dimensional Keller-Segel system with singular sensitivity and signal absorption, J. Differ. Equ., 264 (2018), 2310-2350.  doi: 10.1016/j.jde.2017.10.029.

[29]

M. Winkler, Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differ. Equ., 257 (2014), 1056-1077.  doi: 10.1016/j.jde.2014.04.023.

[30]

Y. Zeng, Nonlinear stability of diffusive contact wave for a chemotaxis model, J. Differ. Equ., 308 (2022), 286-326.  doi: 10.1016/j.jde.2021.11.008.

[31]

Y. Zeng and K. Zhao, On the logarithmic Keller-Segel-Fisher/KPP system, Discrete Contin. Dyn. Syst., 39 (2019), 5365-5402.  doi: 10.3934/dcds.2019220.

[32]

Y. Zeng and K. Zhao, Optimal decay rates for a chemotaxis model with logistic growth, logarithmic sensitivity and density-dependent production/consumption rate, J. Differ. Equ., 268 (2020), 1379-1411. 

[33]

X. Zhao and S. Zheng, Global existence and boundedness of solutions to a chemotaxis system with singular sensitivity and logistic-type source, J. Differ. Equ., 267 (2019), 826-865.  doi: 10.1016/j.jde.2019.01.026.

Figure 1.  The invariant region $ \Omega $
Figure 2.  an invariant region of (13)
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