Structure of solutions to Keller-Segel system

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T.Nagi, Y.Naito and K.Yoshida studied the Keller-Segel system which is the mathematical model describing chemotactic aggregation of cellular slime molds which move toward high cocentrations of chemical substance. Naito and Yoshida with N.Muramoto studied the self-similar solution to the Keller-Segel system. Then the Keller-Segel system is reduced to an elliptic equation with an parameter σ, and obtained solutions of two types, one of which is in a low critical lebel, the other is in a high cirtical lebel. When the parameter is large, there is no self-similar radial solution. This means two solutions are connected by a global blanch. Nagai with T.Senba and T.Suzuki considered the location of blow-up points. These results are announced at each conference or workshop e.g, IMS Workshop on Reaction-Diffusion Systems at HongKong (December 6-10, 1999), The Third World Congress of Nonlinear Analysis at Catania (July 19-26, 2000). H.Usami treated the oscillation problem to the second order ordinary differential equations and the asymptotic properties of the variational eigenvalues. Y.Mizuta completed the results by Koskela concerning the uniqueness property for Sobolev functions.