Chemotactic collapse for the Keller-Segel model |
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Authors: | Miguel A Herrero Juan J L Velázquez |
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Institution: | (1) Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, E-28040 Madrid, Spain, ES |
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Abstract: | This work is concerned with the system
(S) {u
t
=Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v
t
=Δv+(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.
Received 9 March 1995; received in revised form 25 December 1995 |
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Keywords: | AMS (MOS) Subject Classification: 35B55 35B40 35K57 93B05 |
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