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

从1989年5月到1991年3月，对非线性偏微分方程进行了多方面的研究。Masuda首先成功地证明了一类最重要的非线性抛物型方程——反应扩散系统的稳定周期解的存在性。他还考虑了形式为u=*u+a (x) u^p的热方程，并表明如果a (x)在所考虑的区域的某一点上为正，则任何非负解在有限时间内都会爆炸。Matano考虑一维空间范围内具有Dirichlet边界条件的非线性热方程。他可以证明一个显著的结果，即任何具有连续函数作为初始数据的解在（最多）有限个数的点上爆炸。Ishimura证明了*u=u^p形式的非线性椭圆方程的多值解的存在性和不存在性。寿基对永进波进行了研究，分析了永进波的分岔。深谷对微分几何中的非线性微分方程进行了研究。他阐明了截面曲率绝对值小于1的黎曼流形中小球的结构。片冈成功地证明了退化椭圆方程的亚椭圆性。

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.