Title: Global behaviour of radially symmetric solutions stable at infinity for gradient systems,
Abstract: This paper is concerned with radially symmetric solutions of systems of the
form \[ u_t = -\nabla V(u) + \Delta_x u \] where space variable $x$ and and
state-parameter $u$ are multidimensional, and the potential $V$ is coercive at
infinity. For such systems, under generic assumptions on the potential, the
asymptotic behaviour of solutions "stable at infinity", that is approaching a
spatially homogeneous equilibrium when $|x|$ approaches $+\infty$, is
investigated. It is proved that every such solutions approaches a stacked
family of radially symmetric bistable fronts travelling to infinity. This
behaviour is similar to the one of bistable solutions for gradient systems in
one unbounded spatial dimension, described in a companion paper. It is expected
(but unfortunately not proved at this stage) that behind these travelling
fronts the solution again behaves as in the one-dimensional case (that is, the
time derivative approaches zero and the solution approaches a pattern of
stationary solutions).