A blow-up mechanism for a chemotaxis model

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.