Diffusion, attraction and collapse

Michael P Brenner, Peter Constantin, Leo P Kadanoff, Alain Schenkel, Shankar C Venkataramani
1999 Nonlinearity  
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 the possible behaviours of the solutions include convergence to time-independent solutions and the formation of finite-time singularities. Our goal is to describe the different kinds of solutions that lead to these outcomes. We restrict our attention to radial solutions and
more » ... nd 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 stable steady state. On unbounded domains, there exists a one-parameter family of unstable steady solutions and a countable number of unstable blowup behaviours. We document connections between one unstable blowup behaviour and both a stable steady state and a stable blowup, as well as connections between one unstable blowup and two different stable blowups. There is a topological and stability correspondence between the various asymptotic behaviours and this suggests the possibility of constructing a global phase portrait for the system that treats the global in time solutions and the blowing up solutions on an equal footing.
doi:10.1088/0951-7715/12/4/320 fatcat:lgoa2zzzxvhepdsk5ysreiqnj4