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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.

We consider the following system: (S) u t = u − χ∇(u∇v) χ > 0 v = 1 − u which has been used as a model for various phenomena, including motion of species by chemotaxis and equilibrium of self-attracting clusters. We show that, in space dimension N = 3, (S) possess radial solutions that blow-up in a finite time. The asymptotic behaviour of such solutions is analysed in detail. In particular, we obtain that the profile of any such solution consists of an imploding, smoothed-out shock wave that collapses into a Dirac mass when the singularity is formed. The differences between this type of behaviour and that known to occur for blowing-up solutions of (S) in the case N = 2 are also discussed.