  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.
