Structure of solutions to the differential equations arising in natural phenomina

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K.Yoshida, T.Senba and T.Nagi 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. K.Yoshida with Y.Mizutani and N.Muramoto studied the self-similar radial solution to the Keller-Segel system, and obtained the solutions of two types, one of which is in a low critical lebel, the other is in a high cirtical lebel, when a parameter is small. When the parameter is large, there is no self-similar radial solution. These results are extended to the non-radial case, which are prepared. T.Nagai and T.Senba considered the location of blow-up points, whose results are joint work with T.Suzuki and are announced at the conferances at R.M.I of Kyoto University ('97/6/23-6/25) and Kyushu University ('99/2/3-2/5). T.Shibata treated the eigenvalue problem to the nonlinear elliptic equations with two parameters and obtained the asymptotic properties of the variational eigenvalues. Y.Mizuta completed the results by Koskela concerning the uniqueness property for Sobolev functions. R.Ikehata derived an optimal decay rate of energy to initial-boundary problem of hyperbolic equations under geometric conditions on the bounbary.