Skip to main content
SpringerLink
Log in
Book cover

Nonlinear Elliptic and Parabolic Problems pp 273–290Cite as

A Chemotaxis Model with Threshold Density and Degenerate Diffusion

  • Chapter

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 64))

Abstract

A quasilinear degenerate parabolic system modelling the chemotactic movement of cells is studied. The system under consideration has a similar structure as the classical Keller-Segel model, but with the following features: there is a threshold value which the density of cells cannot exceed and the flux of cells vanishes when the density of cells reaches this threshold value. Existence and uniqueness of weak solutions are proved. In the one-dimensional case, flat-hump-shaped stationary solutions are constructed.

The second author was supported by Polish KBN grant 2 P03A 03022.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in “Function Spaces, Differential Operators and Nonlinear Analysis”, H. Triebel, H.J. Schmeisser (eds.), Teubner-Texte Math. 133, Teubner, Stuttgart, 1993, pp. 9–126.

    Google Scholar 

  2. T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math. 26 (2001), 280–301.

    Article  MathSciNet  Google Scholar 

  3. D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105 (2003) 103–165.

    MATH  MathSciNet  Google Scholar 

  4. E. Keller and L. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.

    Article  Google Scholar 

  5. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Paris, 1969.

    Google Scholar 

  6. K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Q. 10 (2002), 501–543.

    MathSciNet  Google Scholar 

  7. C.S. Patlak, Random walk with persistence and external bias, Bull. Math. Biol. Biophys. 15 (1953), 311–338.

    MathSciNet  Google Scholar 

  8. J. Simon, Compact sets in the space LP(0,T;B), Ann. Mat. Pura Appl. 146 (1987), 65–96.

    MATH  MathSciNet  Google Scholar 

  9. D. Wrzosek, Global attractor for a chemotaxis model with prevention of overcrowding, Nonlinear Anal. 59 (2004), 1293–1310.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Wrzosek, Long time behaviour of solutions to a chemotaxis model with volume filling effect, Proc. Roy. Soc. Edinburgh Sect. A, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathématiques pour l’Industrie et la Physique CNRS UMR 5640, Université Paul Sabatier-Toulouse 3, 118 route de Narbonne, F-31062, Toulouse cedex 4, France

    Philippe Laurençot

  2. Institute of Applied Mathematics and Mechanics, Warsaw University, Banacha 2, 02-097, Warszawa, Poland

    Dariusz Wrzosek

Authors
  1. Philippe Laurençot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dariusz Wrzosek
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Universitá Pierre et Marie Curie, Paris

    Haim Brezis

  2. Rutgers University, New Brunswick, N.J.

    Haim Brezis

  3. Angewandte Mathematik, Universität Zurich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Michel Chipot

  4. Institute of Applied Mathematics, University of Hannover, Welfengarten 1, 30167, Hannover, Germany

    Joachim Escher

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Birkhäuser Verlag Basel/Switzerland

About this chapter

Cite this chapter

Laurençot, P., Wrzosek, D. (2005). A Chemotaxis Model with Threshold Density and Degenerate Diffusion. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_16

Download citation

Publish with us

Policies and ethics

Search

Navigation