Diffusion, attraction and collapse

We study a parabolic-elliptic system of partial differential equations that arises in modelling the overdamped gravitational interaction of a cloud of particles or chemotaxis in bacteria. The system has a rich dynamics and th e possible behaviours of the solutions include convergence to time-independ ent solutions and the formation of finite-time singularities. Our goal is t o describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and find that the behaviour of the system depends strongly on the space dimension d. For 2 < d < 10 there are two stable blowup modalities (self-similar and Burgers-like) and one st able steady state. On unbounded domains, there exists a one-parameter famil y of unstable steady solutions and a countable number of unstable blowup be haviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections bet ween one unstable blowup and two different stable blowups. There is a topol ogical and stability correspondence between the various asymptotic behaviou rs and this suggests the possibility of constructing a global phase portrai t for the system that treats the global in time solutions and the blowing u p solutions on an equal footing.