Study of Nonlinear Partial Differential Equations from the View Point of Functional Analysis

在1989年5月至1991年3月期间，我们对非线性偏微分方程进行了多方面的研究。增田首先成功地证明了某些类型的反应扩散系统的稳定周期解的存在性，这是最重要的一类非线性抛物方程。他还考虑了热方程的形式u=*u+a（x）u^p，并表明任何非负解爆破在有限时间内，如果a（x）是积极的在某些点上的域考虑。Matano研究了一维空间区间上的非线性热方程的Dirichlet边值问题，他证明了任何以连续函数为初值的解在有限个点处爆破的显著结果，Ishimura证明了非线性椭圆型方程的多值解的存在性和不存在性。p.shouji对永久前进波进行了研究，并对前进波的分叉进行了分析。福谷研究非线性微分方程微分几何。他澄清了结构的小球在黎曼流形的截面曲率小于一个绝对值。片冈成功地表明亚椭圆退化椭圆方程。

We have studied nonlinear partial differential equations from many aspects in the period 1989 May-1991 March. Masuda succeeded first in showing the existence of stable periodic solutions of reaction-diffusion systems of some type, the most important class of nonlinear parabolic equations. He also considered the heat equation in the form u=*u+a (x) u^p and showed any non-negative solution blows up in a finite time if a (x) is posive at some point in the domain considered. Matano considered the nonlinear heat equation in some one-dimensional space interval with Dirichlet boundary condition.He could show the remarkable result that any solutions with a continuous function as the initial data blows up at a (at most) finite number of points.Ishimura showed the existence and nonexistence of multi-valued solutions of nonlinear elliptic equations of the form *u=u^p. shouji made research into the permanent progressive waves and analyzed the bifurcation of progressive waves. Fukaya made research into nonlinear differential equations in differential geometry. He clarified the structure of small ball in the Riemannian manifold with sectional curvature less than one in absolute value.Kataoka succeeded in showing the hypoellipticity of degenerate elliptic equations.