On the behavior of solutions for nonlinear parabolic or elliptic equations

由于Giga和Kohn的著名结果与Cauchy问题或Dirichlet-Cauchy问题有很大的不同，在有界区域上具有Neumann边界条件的幂非线性半线性热方程解的Blow-up问题似乎还没有结果.我们证明了解的爆破速率和行为与Giga和Kohn与名古屋大学的Kazuhiro Ishige共同工作的结果相同。此外，我们还证明了爆破只发生在具有努曼边界条件的拉普拉斯算子的第二本征函数的极大值点附近。另一方面，上述方程Cauchy问题的整体解的结构在某个指数附近发生突变，Yanagida得到了稳定解的整体稳定性。作为一个应用程序，Yamada研究了de布兰日的线性压缩的互补空间的一些基本性质，并得到了奈良大学的Takahashi关于单位球情形下的扩展插值问题的一个简单证明。Kubota研究了Banach空间中自同构的稳定域。Ito试图应用上述结果研究到实用信息教育。

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.