Mathematical Analysis for Partial Differential Equations Arising in Mathematical Sciences

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We have extensive developments in analytic thoarees on the equations of magretoencaphalography and chamotaxis, For the system of sell-interacting particles we have made the relations clearer between the stochastic differential equation with the terms of friction and fluctuations, (that is, the Langerin equation) the Fokker-Planck equation under the treatment of statistical mechanics, (the is, the kinetic model) parabolic-elliptic system of chemolakis, semilinear elliptic boundary value problems with the exponential nonlinearity, and the system of gradiont flows. We have showed that those systems are to be indicated as the quantization of blow-up mechanism. In particular, we have proven analytically that the collapses are quantiged in the system of chemataxis.