On the behavior of solutions for nonlinear parabolic or elliptic equations

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It seems that there have been no results on the blow-up of solutions for a semilinear heat equation with power nonlinearity under the Neumann boundary condition in a bounded domain since the well-known result by Giga and Kohn cannot work effectively, which is a quite difference from the Cauchy or the Dirichlet-Cauchy problem. I proved that the blow-up rate and behaviors of solutions are as same as that in the result by Giga and Kohn by a joint work with Kazuhiro Ishige of Nagoya University. Moreover, we showed that the bllow-up occurs only near the maximum points of the second eigenfunctions of the laplacian with the Nuemann boundary condition. This result is closely related to the hot spots conjecture for the heat equation.On the other hand, the structure of global solutions of the Cauchy problem for tie above equation changes suddenly near some exponent Yanagida obtained the global stability of stationary solutions. As a application, he showed the existence of unbounded global solutions which behave very complicatedly.Yamada investigated some fundamental properties about the complementary spaces on the linear contraction of de Branges and obtained a simple proof the problem on the extended interpolation in the case of the unit ball due to Takahashi of Nara University.Kubota studied about the stable domain of automorphism in a Banach space.Ito tried to apply the above studies to practical information education.