American Institute of Mathematical Sciences

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June  2015, 20(4): 1231-1250. doi: 10.3934/dcdsb.2015.20.1231

Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity

Qi Wang 1,

1. 

Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130

Received  May 2014 Revised  November 2014 Published  February 2015

In this paper, we study the nonconstant positive steady states of a Keller-Segel chemotaxis system over a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq 1$. The sensitivity function is chosen to be $\phi(v)=\ln (v+c)$ where $c$ is a positive constant. For the chemical diffusion rate being small, we construct positive solutions with a boundary spike supported on a platform. Moreover, this spike approaches the most curved part of the boundary of the domain as the chemical diffusion rate shrinks to zero. We also conduct extensive numerical simulations to illustrate the formation of stable boundary and interior spikes of the system. These spiky solutions can be used to model the self--organized cell aggregation phenomenon in chemotaxis.
Keywords: boundary spike., Chemotaxis, steady-state.
Mathematics Subject Classification: Primary: 92C17, 35K51; Secondary: 35K5.
Citation: Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual Variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

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[4]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Model, 5 (2012), 51.  doi: 10.3934/krm.2012.5.51.  Google Scholar

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[6]

W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, Archive of Rational Mechanics and Analysis, 91 (1986), 283.  doi: 10.1007/BF00282336.  Google Scholar

[7]

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[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$,, Advances in Mathematics, 7A (1981), 369.   Google Scholar

[9]

M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory,, Cal. Var. PDE, 11 (2000), 143.  doi: 10.1007/PL00009907.  Google Scholar

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C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,, Can. J. Math, 52 (2000), 522.  doi: 10.4153/CJM-2000-024-x.  Google Scholar

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T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis,, Journal of Mathematical Biology, 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

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T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius,, Discrete Contin. Dyn. Syst-Series B, 7 (2007), 125.  doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

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E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability,, Journal of Theoratical Biology, 26 (1970), 399.   Google Scholar

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E. F. Keller and L. A. Segel, Model for chemotaxis,, Journal of Theoratical Biology, 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

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E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis,, Journal of Theoratical Biology, 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

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M. K. Kwong and L. Zhang, Uniqueness of positive solutions $\Delta u+f(u)=0$ in an annulus,, Differential and Intergral Equations, 4 (1991), 583.   Google Scholar

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C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system,, Journal of Differential Equation, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

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R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, Journal of Mathematical Biology, 61 (2010), 739.  doi: 10.1007/s00285-009-0317-0.  Google Scholar

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T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis,, Adv. Math. Soc. Appl, 8 (1998), 145.   Google Scholar

[23]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar

[24]

T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis,, RIMS Kokyuroku, 1009 (1997), 22.   Google Scholar

[25]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, Journal. Theor. Biol, 42 (1973), 63.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[26]

W.-M. Ni, Diffusion, cross-diffusion, and their spike layer steady states,, Notices of AMS, 45 (1998), 9.   Google Scholar

[27]

W.-M. Ni, The Mathematics of Diffusion,, CBMS-NSF Regional. Conf. Ser. Appl. Math. 82. SIAM. Philadelphia, 82 (2011).  doi: 10.1137/1.9781611971972.  Google Scholar

[28]

W.-M. Ni and I. Takagi, On the shape of least enery solutions to a semilinear Neumann problem,, Communication of Pure and Applied Math, 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar

[29]

W.-M. Ni and I. Takagi, Location of the peaks of least energy solutions to a semilinear Neumann problem,, Duke Math Journal, 70 (1993), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[30]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial Ekvac, 44 (2001), 441.   Google Scholar

[31]

R. Schaaf, Stationary solutions of chemotaxis systems,, Trans. Amer. Math. Soc, 292 (1985), 531.  doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[32]

B. D. Sleeman, M. J. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model,, SIAM., 65 (2005), 790.  doi: 10.1137/S0036139902415117.  Google Scholar

[33]

W. Strauss, Existence of solitary waves in higher dimensions,, Communications in Mathematical Physics, 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[34]

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

[35]

Q. Wang, Global solutions of a Keller-Segel system with saturated logarithmic sensitivity function,, Commun. Pure Appl. Anal., 14 (2015), 383.   Google Scholar

[36]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics,, SIAM, 31 (2000), 535.  doi: 10.1137/S0036141098339897.  Google Scholar

[37]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem,, Journal of Math. Biol, 66 (2013), 1241.  doi: 10.1007/s00285-012-0533-x.  Google Scholar

[38]

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

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M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, Journal of Differential Equations, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[40]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Mathematical Methods in the Applied Sciences, 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual Variational methods in critical point theory and applications,, J. Functional Analysis, 14 (1973), 349.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar

[2]

M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis,, Bioessays, 28 (2006), 9.  doi: 10.1002/bies.20343.  Google Scholar

[3]

P. Biler, Global solutions to some parabolic elliptic systems of chemotaxis,, Adv. Math. Sci. Appl., 9 (1999), 347.   Google Scholar

[4]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Model, 5 (2012), 51.  doi: 10.3934/krm.2012.5.51.  Google Scholar

[5]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Bioscience, 56 (1981), 217.  doi: 10.1016/0025-5564(81)90055-9.  Google Scholar

[6]

W.-Y. Ding and W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, Archive of Rational Mechanics and Analysis, 91 (1986), 283.  doi: 10.1007/BF00282336.  Google Scholar

[7]

D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation,, Current Opinion in Genetics Development, 16 (2006), 367.  doi: 10.1016/j.gde.2006.06.003.  Google Scholar

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^N$,, Advances in Mathematics, 7A (1981), 369.   Google Scholar

[9]

M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory,, Cal. Var. PDE, 11 (2000), 143.  doi: 10.1007/PL00009907.  Google Scholar

[10]

C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,, Can. J. Math, 52 (2000), 522.  doi: 10.4153/CJM-2000-024-x.  Google Scholar

[11]

M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model,, Journal of Mathematical Biology, 35 (1996), 177.  doi: 10.1007/s002850050049.  Google Scholar

[12]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis,, Journal of Mathematical Biology, 58 (2009), 183.  doi: 10.1007/s00285-008-0201-3.  Google Scholar

[13]

T. Hillen, K. J. Painter and C. Schmeiser, Global existence for Chemotaxis with finite sampling radius,, Discrete Contin. Dyn. Syst-Series B, 7 (2007), 125.  doi: 10.3934/dcdsb.2007.7.125.  Google Scholar

[14]

D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences I,, Jahresber DMV, 105 (2003), 103.   Google Scholar

[15]

D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences II,, Jahresber DMV, 106 (2004), 51.   Google Scholar

[16]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability,, Journal of Theoratical Biology, 26 (1970), 399.   Google Scholar

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis,, Journal of Theoratical Biology, 30 (1971), 225.  doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[18]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis,, Journal of Theoratical Biology, 30 (1971), 235.  doi: 10.1016/0022-5193(71)90051-8.  Google Scholar

[19]

M. K. Kwong and L. Zhang, Uniqueness of positive solutions $\Delta u+f(u)=0$ in an annulus,, Differential and Intergral Equations, 4 (1991), 583.   Google Scholar

[20]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system,, Journal of Differential Equation, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[21]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models,, Journal of Mathematical Biology, 61 (2010), 739.  doi: 10.1007/s00285-009-0317-0.  Google Scholar

[22]

T. Nagai and T. Senba, Global existence and blow-up of radial solutions to a parabolic-elliptic system of chemotaxis,, Adv. Math. Soc. Appl, 8 (1998), 145.   Google Scholar

[23]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411.   Google Scholar

[24]

T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis,, RIMS Kokyuroku, 1009 (1997), 22.   Google Scholar

[25]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology,, Journal. Theor. Biol, 42 (1973), 63.  doi: 10.1016/0022-5193(73)90149-5.  Google Scholar

[26]

W.-M. Ni, Diffusion, cross-diffusion, and their spike layer steady states,, Notices of AMS, 45 (1998), 9.   Google Scholar

[27]

W.-M. Ni, The Mathematics of Diffusion,, CBMS-NSF Regional. Conf. Ser. Appl. Math. 82. SIAM. Philadelphia, 82 (2011).  doi: 10.1137/1.9781611971972.  Google Scholar

[28]

W.-M. Ni and I. Takagi, On the shape of least enery solutions to a semilinear Neumann problem,, Communication of Pure and Applied Math, 44 (1991), 819.  doi: 10.1002/cpa.3160440705.  Google Scholar

[29]

W.-M. Ni and I. Takagi, Location of the peaks of least energy solutions to a semilinear Neumann problem,, Duke Math Journal, 70 (1993), 247.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[30]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations,, Funkcial Ekvac, 44 (2001), 441.   Google Scholar

[31]

R. Schaaf, Stationary solutions of chemotaxis systems,, Trans. Amer. Math. Soc, 292 (1985), 531.  doi: 10.1090/S0002-9947-1985-0808736-1.  Google Scholar

[32]

B. D. Sleeman, M. J. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model,, SIAM., 65 (2005), 790.  doi: 10.1137/S0036139902415117.  Google Scholar

[33]

W. Strauss, Existence of solitary waves in higher dimensions,, Communications in Mathematical Physics, 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[34]

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

[35]

Q. Wang, Global solutions of a Keller-Segel system with saturated logarithmic sensitivity function,, Commun. Pure Appl. Anal., 14 (2015), 383.   Google Scholar

[36]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics,, SIAM, 31 (2000), 535.  doi: 10.1137/S0036141098339897.  Google Scholar

[37]

X. Wang and Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly's compactness theorem,, Journal of Math. Biol, 66 (2013), 1241.  doi: 10.1007/s00285-012-0533-x.  Google Scholar

[38]

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

[39]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model,, Journal of Differential Equations, 248 (2010), 2889.  doi: 10.1016/j.jde.2010.02.008.  Google Scholar

[40]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity,, Mathematical Methods in the Applied Sciences, 34 (2011), 176.  doi: 10.1002/mma.1346.  Google Scholar

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