A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in ${\mathbb{R}}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.

半线性椭圆偏微分方程（PDE）在有限域上的刻画 ${\mathbb{[R}}^{ñ}$根据无限维动力系统给出。动态系统位于PDE的边界数据空间中。这是解决椭圆问题的一种新颖方法，可以在研究相应的PDE时使用动力学系统工具。动态系统是不适当的，这意味着对于通用初始数据，解决方案在时间上不会向前或向后存在。我们提供了一个可以分析此不适系统的框架。这可以看作是对空间动力学理论的概括，适用于无限圆柱域的情况。