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A blow-up mechanism for a chemotaxis model

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Herrero, Miguel A. and Velázquez, J.J. L. (1997) A blow-up mechanism for a chemotaxis model. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 24 (4). pp. 633-683. ISSN 0391-173X

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Official URL: http://www.numdam.org/item?id=ASNSP_1997_4_24_4_633_0


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Abstract

We consider the following nonlinear system of parabolic equations: (1) ut =Δu−χ∇(u∇v), Γvt =Δv+u−av for x∈B R, t>0. Here Γ,χ and a are positive constants and BR is a ball of radius R>0 in R2. At the boundary of BR, we impose homogeneous Neumann conditions, namely: (2) ∂u/∂n=∂v/∂n=0 for |x|=R, t>0.
Problem (1),(2) is a classical model to describe chemotaxis, i.e., the motion of organisms induced by high concentrations of a chemical that they secrete. In this paper we prove that there exist radial solutions of (1),(2) that develop a Dirac-delta type singularity in finite time, a feature known in the literature as chemotactic collapse. The asymptotics of such solutions near the formation of the singularity is described in detail, and particular attention is paid to the structure of the inner layer around the unfolding singularity.


Item Type:Article
Uncontrolled Keywords:Homogeneous Neumann conditions; Dirac-delta type singularity in finite time; inner layer around the unfolding singularity
Subjects:Medical sciences > Biology > Biomathematics
Sciences > Mathematics > Differential equations
ID Code:22658
Deposited On:27 Aug 2013 07:51
Last Modified:12 Dec 2018 15:08

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