Journal of Mathematical Biology

, Volume 35, Issue 2, pp 177–194

Chemotactic collapse for the Keller-Segel model

  • Miguel A. Herrero
  • Juan J. L. Velázquez

Abstract.

 This work is concerned with the system

(S) {utu − χ∇ (uv) for x∈Ω, t>0Γ vtv+(u−1) for x∈Ω, t>0

where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ2. On the boundary ∂Ω, we impose no-flux conditions:

(N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0

Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species.

AMS (MOS) Subject Classification: 35B55 35B40 35K57 93B05. 
Key words: Chemotaxis Advection-diffusion systems Matched asymptotic expansions Blow-up Asymptotic behaviour 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Miguel A. Herrero
    • 1
  • Juan J. L. Velázquez
    • 1
  1. 1.Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, E-28040 Madrid, SpainES

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