Chemotactic collapse for the Keller-Segel model

This work is concerned with the system (S) {u(t)=Delta u-chi del(u del upsilon) for x is an element of Omega, t>0 Gamma upsilon(t)=Delta upsilon=Delta upsilon+(u-1) for x is an element of Omega, t>0 where Gamma; chi are positive constants and Omega is a bounded and smooth open set in IR(2). On the boundary delta Omega, we impose no-flux conditions: (N)partial derivative u/partial derivative n=partial derivative upsilon/partial derivative n=0 for x is an element of partial derivative Omega, 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), upsilon(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.