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October  2018, 11(5): 1085-1123. doi: 10.3934/krm.2018042

Boundary layers and stabilization of the singular Keller-Segel system

1. 

Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China

2. 

Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

3. 

Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

4. 

School of Mathematics, South China University of Technology, Guangzhou 510641, China

* Corresponding author

Received  June 2017 Revised  September 2017 Published  May 2018

The original Keller-Segel system proposed in [23] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter ε now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as ε→0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in-ε estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L1-estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

Citation: Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042
References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. Google Scholar

[2]

W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modeling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691-722. doi: 10.1007/BF00275511. Google Scholar

[3]

D. Balding and D. L. S. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol., 114 (1985), 53-73. doi: 10.1016/S0022-5193(85)80255-1. Google Scholar

[4]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford University Press, 2000. Google Scholar

[5]

M. Chae, K. Choi, K. Kang and J. Lee, Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain, Journal of Differential Equations, 265 (2018), 237-279, arXiv: 1609.00821v1. doi: 10.1016/j.jde.2018.02.034. Google Scholar

[6]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168. Google Scholar

[7]

L. CorriasB. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 2 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0. Google Scholar

[8]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar

[9]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th Edition, Spring-Verlag, 2016. doi: 10.1007/978-3-662-49451-6. Google Scholar

[10]

F. W. DahlquistP. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123. Google Scholar

[11]

P. N. Davis, P. van Heijster and R. Marangell, Absolute instabilities of traveling wave solutions in a Keller-Segel model, arXiv: 1608.05480v2.Google Scholar

[12]

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. Google Scholar

[13]

H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids, Comm. Math. Phys., 208 (1999), 309-330. doi: 10.1007/s002200050760. Google Scholar

[14]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101. Google Scholar

[15]

C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew Math. Phys., 63 (2012), 825-834. doi: 10.1007/s00033-012-0193-0. Google Scholar

[16]

H. HöferJ. A. Sherratt and P. K. Maini, Cellular pattern formation during Dictyostelium aggregation, Physica D., 85 (1995), 425-444. Google Scholar

[17]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111. Google Scholar

[18]

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

[19]

S. Jiang and J. W. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612. Google Scholar

[20]

S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268. doi: 10.1137/07070005X. Google Scholar

[21]

H. Y. JinJ. Y. 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. Google Scholar

[22]

Y. V. KalininL. JiangY. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448. doi: 10.1016/j.bpj.2008.10.027. Google Scholar

[23]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[24]

E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1976), 309-317. doi: 10.1016/0025-5564(75)90109-1. Google Scholar

[25]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[26]

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. Google Scholar

[27]

H. A. LevineB. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. i. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 77-115. doi: 10.1016/S0025-5564(00)00034-1. Google Scholar

[28]

D. LiR. Pan and K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 7 (2015), 2181-2210. doi: 10.1088/0951-7715/28/7/2181. Google Scholar

[29]

H. Li and K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338. doi: 10.1016/j.jde.2014.09.014. Google Scholar

[30]

J. LiT. Li and Z. A. Wang, Stability of traveling waves of the keller-segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849. doi: 10.1142/S0218202514500389. Google Scholar

[31]

T. LiR. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443. doi: 10.1137/110829453. Google Scholar

[32]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. doi: 10.1137/09075161X. Google Scholar

[33]

T. Li and Z. A. Wang, Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998. doi: 10.1142/S0218202510004830. Google Scholar

[34]

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

[35]

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

[36]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. Ⅱ, Compressible Models, Clarendon Press, 1998. Google Scholar

[37]

V. 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., to appear.Google Scholar

[38]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, 3rd edition, Springer, Berlin, 2002. Google Scholar

[39]

T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334. Google Scholar

[40]

R. Nossal, Boundary movement of chemotactic bacterial populations, Math. Biosci., 13 (1972), 397-406. doi: 10.1016/0025-5564(72)90058-2. Google Scholar

[41]

K. J. PainterP. K. Maini and H. G. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis, Proc. Natl. Acad. Sci., 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549. Google Scholar

[42]

K. J. PainterP. K. Maini and H. G. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bull. Math. Biol., 62 (2000), 501-525. Google Scholar

[43]

H. Y. PengH. Y. Wen and C. J. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew Math. Phys, 65 (2014), 1167-1188. doi: 10.1007/s00033-013-0378-1. Google Scholar

[44]

G. J. PetterH. M. ByrneD. L. S. McElwain and J. Norbury, A model of wound healing and angiogenesis in soft tissue, Math. Biosci., 136 (2003), 35-63. Google Scholar

[45]

X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal, 216 (2015), 1049-1086. doi: 10.1007/s00205-014-0826-x. Google Scholar

[46]

H. Schwetlick, Traveling waves for chemotaxis systems, Prof. Appl. Math. Mech., 3 (2003), 476-478. doi: 10.1002/pamm.200310508. Google Scholar

[47]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd Edition, Spring-Verlag, Berlin, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[48]

R. TysonS. R. Lubkin and J. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375. doi: 10.1007/s002850050153. Google Scholar

[49]

Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst - B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. Google Scholar

[50]

Z. A. WangY. S. Tao and L. H. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. System - B, 18 (2013), 821-845. Google Scholar

[51]

Z. A. WangZ. 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. Google Scholar

[52]

Z. A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Comm. Pure Appl. Anal., 12 (2013), 3027-3046. doi: 10.3934/cpaa.2013.12.3027. Google Scholar

[53]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024. doi: 10.1142/S0218202516500238. Google Scholar

[54]

L. YaoT. Zhang and C. J. Zhu, Boundary layers for compressible Navier-Stokes equations with density-dependent viscosity and cylindrical symmetry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 677-709. doi: 10.1016/j.anihpc.2011.04.006. Google Scholar

show all references

References:
[1]

J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. Google Scholar

[2]

W. Alt and D. A. Lauffenburger, Transient behavior of a chemotaxis system modeling certain types of tissue inflammation, J. Math. Biol., 24 (1987), 691-722. doi: 10.1007/BF00275511. Google Scholar

[3]

D. Balding and D. L. S. McElwain, A mathematical model of tumour-induced capillary growth, J. Theor. Biol., 114 (1985), 53-73. doi: 10.1016/S0022-5193(85)80255-1. Google Scholar

[4]

A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford University Press, 2000. Google Scholar

[5]

M. Chae, K. Choi, K. Kang and J. Lee, Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain, Journal of Differential Equations, 265 (2018), 237-279, arXiv: 1609.00821v1. doi: 10.1016/j.jde.2018.02.034. Google Scholar

[6]

M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor, IMA J. Math. Appl. Med., 10 (1993), 149-168. Google Scholar

[7]

L. CorriasB. Perthame and H. Zaag, A chemotaxis model motivated by angiogenesis, C. R. Math. Acad. Sci. Paris, 2 (2003), 141-146. doi: 10.1016/S1631-073X(02)00008-0. Google Scholar

[8]

L. CorriasB. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x. Google Scholar

[9]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th Edition, Spring-Verlag, 2016. doi: 10.1007/978-3-662-49451-6. Google Scholar

[10]

F. W. DahlquistP. Lovely and D. E. Jr Koshland, Qualitative analysis of bacterial migration in chemotaxis, Nature, New Biol., 236 (1972), 120-123. Google Scholar

[11]

P. N. Davis, P. van Heijster and R. Marangell, Absolute instabilities of traveling wave solutions in a Keller-Segel model, arXiv: 1608.05480v2.Google Scholar

[12]

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. Google Scholar

[13]

H. Frid and V. Shelukhin, Boundary layers for the Navier-Stokes equations of compressible fluids, Comm. Math. Phys., 208 (1999), 309-330. doi: 10.1007/s002200050760. Google Scholar

[14]

A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 (2003), 118101. doi: 10.1103/PhysRevLett.90.118101. Google Scholar

[15]

C. Hao, Global well-posedness for a multidimensional chemotaxis model in critical Besov spaces, Z. Angew Math. Phys., 63 (2012), 825-834. doi: 10.1007/s00033-012-0193-0. Google Scholar

[16]

H. HöferJ. A. Sherratt and P. K. Maini, Cellular pattern formation during Dictyostelium aggregation, Physica D., 85 (1995), 425-444. Google Scholar

[17]

D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), 215-254. doi: 10.1006/jdeq.1995.1111. Google Scholar

[18]

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

[19]

S. Jiang and J. W. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics, Nonlinearity, 30 (2017), 3587-3612. Google Scholar

[20]

S. Jiang and J. W. Zhang, Boundary layers for the Navier-Stokes equations of compressible heat-conducting flows with cylindrical symmetry, SIAM J. Math. Anal., 41 (2009), 237-268. doi: 10.1137/07070005X. Google Scholar

[21]

H. Y. JinJ. Y. 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. Google Scholar

[22]

Y. V. KalininL. JiangY. Tu and M. Wu, Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448. doi: 10.1016/j.bpj.2008.10.027. Google Scholar

[23]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8. Google Scholar

[24]

E. F. Keller and G. M. Odell, Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1976), 309-317. doi: 10.1016/0025-5564(75)90109-1. Google Scholar

[25]

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[26]

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. Google Scholar

[27]

H. A. LevineB. D. Sleeman and M. Nilsen-Hamilton, A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. i. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 77-115. doi: 10.1016/S0025-5564(00)00034-1. Google Scholar

[28]

D. LiR. Pan and K. Zhao, Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 7 (2015), 2181-2210. doi: 10.1088/0951-7715/28/7/2181. Google Scholar

[29]

H. Li and K. Zhao, Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338. doi: 10.1016/j.jde.2014.09.014. Google Scholar

[30]

J. LiT. Li and Z. A. Wang, Stability of traveling waves of the keller-segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849. doi: 10.1142/S0218202514500389. Google Scholar

[31]

T. LiR. Pan and K. Zhao, Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443. doi: 10.1137/110829453. Google Scholar

[32]

T. Li and Z. A. Wang, Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541. doi: 10.1137/09075161X. Google Scholar

[33]

T. Li and Z. A. Wang, Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis, Math. Models Methods Appl. Sci., 20 (2010), 1967-1998. doi: 10.1142/S0218202510004830. Google Scholar

[34]

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

[35]

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

[36]

P. L. Lions, Mathematical Topics in Fluid Mechanics. Vol. Ⅱ, Compressible Models, Clarendon Press, 1998. Google Scholar

[37]

V. 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., to appear.Google Scholar

[38]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, 3rd edition, Springer, Berlin, 2002. Google Scholar

[39]

T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184. doi: 10.1007/BF00160334. Google Scholar

[40]

R. Nossal, Boundary movement of chemotactic bacterial populations, Math. Biosci., 13 (1972), 397-406. doi: 10.1016/0025-5564(72)90058-2. Google Scholar

[41]

K. J. PainterP. K. Maini and H. G. Othmer, Stripe formation in juvenile pomacanthus explained by a generalized Turing mechanism with chemotaxis, Proc. Natl. Acad. Sci., 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549. Google Scholar

[42]

K. J. PainterP. K. Maini and H. G. Othmer, A chemotactic model for the advance and retreat of the primitive streak in avian development, Bull. Math. Biol., 62 (2000), 501-525. Google Scholar

[43]

H. Y. PengH. Y. Wen and C. J. Zhu, Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew Math. Phys, 65 (2014), 1167-1188. doi: 10.1007/s00033-013-0378-1. Google Scholar

[44]

G. J. PetterH. M. ByrneD. L. S. McElwain and J. Norbury, A model of wound healing and angiogenesis in soft tissue, Math. Biosci., 136 (2003), 35-63. Google Scholar

[45]

X. L. QinT. YangZ. A. Yao and W. S. Zhou, Vanishing shear viscosity and boundary layer for the Navier-Stokes equations with cylindrical symmetry, Arch. Ration. Mech. Anal, 216 (2015), 1049-1086. doi: 10.1007/s00205-014-0826-x. Google Scholar

[46]

H. Schwetlick, Traveling waves for chemotaxis systems, Prof. Appl. Math. Mech., 3 (2003), 476-478. doi: 10.1002/pamm.200310508. Google Scholar

[47]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd Edition, Spring-Verlag, Berlin, 1994. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[48]

R. TysonS. R. Lubkin and J. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375. doi: 10.1007/s002850050153. Google Scholar

[49]

Z. A. Wang, Mathematics of traveling waves in chemotaxis, Discrete Contin. Dyn. Syst - B, 18 (2013), 601-641. doi: 10.3934/dcdsb.2013.18.601. Google Scholar

[50]

Z. A. WangY. S. Tao and L. H. Wang, Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension, Discrete Contin. Dyn. System - B, 18 (2013), 821-845. Google Scholar

[51]

Z. A. WangZ. 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. Google Scholar

[52]

Z. A. Wang and K. Zhao, Global dynamics and diffusion limit of a one-dimensional repulsive chemotaxis model, Comm. Pure Appl. Anal., 12 (2013), 3027-3046. doi: 10.3934/cpaa.2013.12.3027. Google Scholar

[53]

M. Winkler, The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties, Math. Models Methods Appl. Sci., 26 (2016), 987-1024. doi: 10.1142/S0218202516500238. Google Scholar

[54]

L. YaoT. Zhang and C. J. Zhu, Boundary layers for compressible Navier-Stokes equations with density-dependent viscosity and cylindrical symmetry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 677-709. doi: 10.1016/j.anihpc.2011.04.006. Google Scholar

Figure 1.  Numerical simulation of the evolution of solution profiles of the system (4) as $\varepsilon\to 0$ in the interval $[0, 1]$, where $u|_{x=0, 1}=1+0.1\sin(t), v|_{x=0, 1}=1+0.1\sin(t), u_0(x)=1-\sin(\pi x), v_0(x)=1+x(1-x)$. The solution $(u(x,t), v(x,t)$ is plotted at time $t=0.2$
Figure 2.  Numerical simulation of the time evolution of boundary layer solutions of (4) with $\varepsilon=0.0001$ in the interval $[0, 1]$, where the initial and boundary date are same as those chosen in Fig. 1
Figure 3.  Numerical simulation of the time evolution of solutions to (4) in the interval $[0, 1]$ with decay boundary data, where $u|_{x=0,1}=1+\exp(-t), v|_{x=0, 1}=1/(1+t), u_0(x)=2+x(1-x),v_0(x)=1+x(1-x)$, and $\chi=D=1, \varepsilon=0.0001$
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