Asymptotic self-similar blow-up for a model of aggregation

Abstract

In this article we consider a system of equations that describes a class of massconserving 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. We compare these results with blow-up behaviour of related models of aggregation.2000 Mathematics Subject Classification: 35Q, 35K60, 35B40, 82C21. Key words and phrases: aggregation of particles models, nonlinear nonlocal parabolic system, finite time blow-up of solutions.The paper is in final form and no version of it will be published elsewhere.[165]