Chemotactic collapse for the Keller-Segel model

This work is concerned with the system (equation: see text), where Gamma, chi are positive constants and Omega is a bounded and smooth open set in IR2. On the boundary delta Omega, we impose no-flux conditions: (equation: see text). 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) --> A delta (y) as t --> T for some T < infinity, where A is the total concentration of the species.