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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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

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

[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.

[38]

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

[39]

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

[40]

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

[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.

[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.

[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.

[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.

[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.

[46]

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

[47]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

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

[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.

[38]

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

[39]

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

[40]

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

[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.

[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.

[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.

[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.

[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.

[46]

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

[47]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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