Singular solutions of a nonlinear elliptic equation and an infinite dimensional dynamical system
Singular solutions of a nonlinear elliptic equation and an infinite dimensional dynamical system
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非线性椭圆方程和无限维动力系统的奇异解
DOI：
10.1007/bfb0084899
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发表时间：
1990
期刊：
影响因子：
--
通讯作者：
H. Matano
中科院分区：
文献类型：
--
作者：
H. Matano
选择其中一个空间变量作为时间变量，可以将某类椭圆型偏微分方程转化为自主时间演化方程。当然，得到的演化方程仍然是椭圆型的，因此是不适定的。尽管如此，在某些情况下，人们可以在动力系统或类似系统的框架中处理问题。请注意，与这些方程相关的动力系统是无限维的，因为这些偏微分方程都不能转换成有限的常微分方程系统。本文讨论了一类非线性椭圆型方程的奇异解，研究了它们在奇点附近和无穷远处的性质。这里使用的主要技术来自无限维动力系统理论。使用动力系统的观点的优点是，人们可以从几何角度洞察问题。由于这种方法，我们在本文中提出的结果具有一定的全局性，并且可能难以通过其他传统的分析方法获得。
Some classes of partial differential equations of elliptic type can be converted into autonomous time-evolution equations by choosing one of the space variables as time variable. The resulting evolution equations are, of course, still of elliptic type, hence are ill-posed. Nonetheless, there are cases in which one can deal with the problem in the framework of dynamical systems or their analogues. Note that the dynamical systems relevant to those equations are of infinite dimensions, since none of those partial differential equations can be converted into a finite system of ordinary differential equations. In this paper we deal with singular solutions of some nonlinear elliptic equation and study their behavior near the singular point and near the infinity. The main techniques used here come from the theory of infinite dimensional dynamical systems. The advantage of using the dynamical systems point of view is that one can get a geometrical insight into the problem. Thanks to this approach, the results we present in this paper are somewhat of global nature and are perhaps difficult to obtain by other conventional methods of analysis.