  It was recently shown by the authors that a semilinear elliptic equation can
be represented as an infinite-dimensional dynamical system in terms of boundary
data on a shrinking one-parameter family of domains. The resulting system is
ill-posed, in the sense that solutions do not typically exist forward or
backward in time. In this paper we consider a radial family of domains and
prove that the linearized system admits an exponential dichotomy, with the
unstable subspace corresponding to the boundary data of weak solutions to the
linear PDE. This generalizes the spatial dynamics approach, which applies to
infinite cylindrical (channel) domains, and also generalizes previous work on
radial domains as we impose no symmetry assumptions on the equation or its
solutions.
