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A blow-up mechanism for a chemotaxis model. (English) Zbl 0904.35037

Summary: We consider the following nonlinear system of parabolic equations: \[ u_t= \Delta u-\chi\nabla(u\nabla v)\quad \Gamma v_t= \Delta v+ u- av\quad \text{for } x\in B_R,\quad t>0.\tag{1} \] Here \(\Gamma\), \(\chi\) and \(a\) are positive constants, and \(B_R\) is a ball of radius \(R>0\) in \(\mathbb{R}^2\). At the boundary of \(B_R\), we impose homogeneous Neumann conditions, namely: \[ {\partial u\over\partial n}= {\partial v\over\partial n}= 0\quad\text{for }| x|= R,\quad t> 0.\tag{2} \] 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.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
35B40 Asymptotic behavior of solutions to PDEs
35A20 Analyticity in context of PDEs
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References:

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