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 & 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

[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

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

[1]

Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks & Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011

[2]

Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147

[3]

Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373

[4]

Tomoyuki Miyaji, Yoshio Tsutsumi. Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1633-1650. doi: 10.3934/cpaa.2018078

[5]

Ken Shirakawa. Stability for steady-state patterns in phase field dynamics associated with total variation energies. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1215-1236. doi: 10.3934/dcds.2006.15.1215

[6]

Wing-Cheong Lo. Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 775-787. doi: 10.3934/dcdsb.2014.19.775

[7]

Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643

[8]

Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135

[9]

Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691

[10]

Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587

[11]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[12]

Christos Sourdis. Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1039-1059. doi: 10.3934/dcds.2017043

[13]

Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046

[14]

Kazuhiro Kurata, Kotaro Morimoto. Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 139-164. doi: 10.3934/dcds.2011.31.139

[15]

Anne Nouri, Christian Schmeiser. Aggregated steady states of a kinetic model for chemotaxis. Kinetic & Related Models, 2017, 10 (1) : 313-327. doi: 10.3934/krm.2017013

[16]

Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613

[17]

Chao Xing, Ping Zhou, Hong Luo. The steady state solutions to thermohaline circulation equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3709-3722. doi: 10.3934/dcdsb.2016117

[18]

Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945

[19]

Orazio Muscato, Wolfgang Wagner, Vincenza Di Stefano. Properties of the steady state distribution of electrons in semiconductors. Kinetic & Related Models, 2011, 4 (3) : 809-829. doi: 10.3934/krm.2011.4.809

[20]

Youcef Amirat, Kamel Hamdache. Steady state solutions of ferrofluid flow models. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2329-2355. doi: 10.3934/cpaa.2016039

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]