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

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.
Citation: Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete and 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), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22. doi: 10.1002/bies.20343.

[3]

P. Biler, Global solutions to some parabolic elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.

[4]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Model, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.

[5]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9.

[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), 283-308. doi: 10.1007/BF00282336.

[7]

D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373. doi: 10.1016/j.gde.2006.06.003.

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N2$, Advances in Mathematics, Supl Study, 7A (1981), 369-402.

[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), 143-175. doi: 10.1007/PL00009907.

[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), 522-538. doi: 10.4153/CJM-2000-024-x.

[11]

M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194. doi: 10.1007/s002850050049.

[12]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[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), 125-144. doi: 10.3934/dcdsb.2007.7.125.

[14]

D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165.

[15]

D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69.

[16]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415.

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[18]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[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), 583-599.

[20]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system, Journal of Differential Equation, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.

[21]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, Journal of Mathematical Biology, 61 (2010), 739-761. doi: 10.1007/s00285-009-0317-0.

[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), 145-156.

[23]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

[24]

T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28.

[25]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol, 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.

[26]

W.-M. Ni, Diffusion, cross-diffusion, and their spike layer steady states, Notices of AMS, 45 (1998), 9-18.

[27]

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

[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), 819-851. doi: 10.1002/cpa.3160440705.

[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), 247-281. doi: 10.1215/S0012-7094-93-07004-4.

[30]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469.

[31]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.

[32]

B. D. Sleeman, M. J. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM., Journal of Applied Math, 65 (2005), 790-817. doi: 10.1137/S0036139902415117.

[33]

W. Strauss, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55 (1977), 149-162. doi: 10.1007/BF01626517.

[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), 821-845.

[35]

Q. Wang, Global solutions of a Keller-Segel system with saturated logarithmic sensitivity function, Commun. Pure Appl. Anal., 14 (2015), 383-396.

[36]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM, Journal of Mathematical Analysis, 31 (2000), 535-560. doi: 10.1137/S0036141098339897.

[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), 1241-1266. doi: 10.1007/s00285-012-0533-x.

[38]

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

[39]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[40]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190. doi: 10.1002/mma.1346.

show all references

References:
[1]

A. Ambrosetti and P. Rabinowitz, Dual Variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7.

[2]

M. D. Baker, P. M. Wolanin and J. B. Stock, Signal transduction in bacterial chemotaxis, Bioessays, 28 (2006), 9-22. doi: 10.1002/bies.20343.

[3]

P. Biler, Global solutions to some parabolic elliptic systems of chemotaxis, Adv. Math. Sci. Appl., 9 (1999), 347-359.

[4]

A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Model, 5 (2012), 51-95. doi: 10.3934/krm.2012.5.51.

[5]

S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Bioscience, 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9.

[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), 283-308. doi: 10.1007/BF00282336.

[7]

D. Dormann and C. Weijer, Chemotactic cell movement during Dictyostelium development and gastrulation, Current Opinion in Genetics Development, 16 (2006), 367-373. doi: 10.1016/j.gde.2006.06.003.

[8]

B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N2$, Advances in Mathematics, Supl Study, 7A (1981), 369-402.

[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), 143-175. doi: 10.1007/PL00009907.

[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), 522-538. doi: 10.4153/CJM-2000-024-x.

[11]

M. A. Herrero and J. J. L. Velazquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194. doi: 10.1007/s002850050049.

[12]

T. Hillen and K. J. Painter, A user's guidence to PDE models for chemotaxis, Journal of Mathematical Biology, 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3.

[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), 125-144. doi: 10.3934/dcdsb.2007.7.125.

[14]

D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences I, Jahresber DMV, 105 (2003), 103-165.

[15]

D. Horstmann, From 1970 until now: The Keller-Segel model in Chemotaxis and its consequences II, Jahresber DMV, 106 (2004), 51-69.

[16]

E. F. Keller and L. A. Segel, Inition of slime mold aggregation view as an instability, Journal of Theoratical Biology, 26 (1970), 399-415.

[17]

E. F. Keller and L. A. Segel, Model for chemotaxis, Journal of Theoratical Biology, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.

[18]

E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A Theretical Analysis, Journal of Theoratical Biology, 30 (1971), 235-248. doi: 10.1016/0022-5193(71)90051-8.

[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), 583-599.

[20]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitute stationary solutions to a chemotaxis system, Journal of Differential Equation, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7.

[21]

R. Lui and Z. A. Wang, Traveling wave solutions from microscopic to macroscopic chemotaxis models, Journal of Mathematical Biology, 61 (2010), 739-761. doi: 10.1007/s00285-009-0317-0.

[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), 145-156.

[23]

T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

[24]

T. Nagai, T. Senba and K. Yoshida, Global existence of solutions to the parabolic systems of chemotaxis, RIMS Kokyuroku, 1009 (1997), 22-28.

[25]

V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology, Journal. Theor. Biol, 42 (1973), 63-105. doi: 10.1016/0022-5193(73)90149-5.

[26]

W.-M. Ni, Diffusion, cross-diffusion, and their spike layer steady states, Notices of AMS, 45 (1998), 9-18.

[27]

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

[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), 819-851. doi: 10.1002/cpa.3160440705.

[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), 247-281. doi: 10.1215/S0012-7094-93-07004-4.

[30]

K. Osaki and A. Yagi, Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial Ekvac, 44 (2001), 441-469.

[31]

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, 292 (1985), 531-556. doi: 10.1090/S0002-9947-1985-0808736-1.

[32]

B. D. Sleeman, M. J. Ward and J. Wei, The existence, stability, and dynamics of spike patterns in a chemotaxis model, SIAM., Journal of Applied Math, 65 (2005), 790-817. doi: 10.1137/S0036139902415117.

[33]

W. Strauss, Existence of solitary waves in higher dimensions, Communications in Mathematical Physics, 55 (1977), 149-162. doi: 10.1007/BF01626517.

[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), 821-845.

[35]

Q. Wang, Global solutions of a Keller-Segel system with saturated logarithmic sensitivity function, Commun. Pure Appl. Anal., 14 (2015), 383-396.

[36]

X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM, Journal of Mathematical Analysis, 31 (2000), 535-560. doi: 10.1137/S0036141098339897.

[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), 1241-1266. doi: 10.1007/s00285-012-0533-x.

[38]

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

[39]

M. Winkler, Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, Journal of Differential Equations, 248 (2010), 2889-2905. doi: 10.1016/j.jde.2010.02.008.

[40]

M. Winkler, Global solutions in a fully parabolic chemotaxis system with singular sensitivity, Mathematical Methods in the Applied Sciences, 34 (2011), 176-190. doi: 10.1002/mma.1346.

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