  In this paper we consider a system of equations that describes a class of
mass-conserving aggregation phenomena, including gravitational collapse and
bacterial chemotaxis. In spatial dimensions strictly larger than two, and under
the assumptions of radial symmetry, it is known that this system has at least
two stable mechanisms of singularity formation (see e.g. M.P. Brenner et
al.1999, Nonlinearity, 12, 1071-1098); one type is self-similar, and may be
viewed as a trade-off between diffusion and attraction, while in the other type
the attraction prevails over the diffusion and a non-self-similar shock wave
results. Our main result identifies a class of initial data for which the
blow-up behaviour is of the former, self-similar type. The blow-up profile is
characterized as belonging to a subset of stationary solutions of the
associated ordinary differential equation.
