  We study the asymptotic (in time) behavior of positive and sign-changing
solutions to nonlinear parabolic problems in the whole space or in the exterior
of a ball with Dirichlet boundary conditions. We show that, under suitable
regularity and stability assumptions, solutions are asymptotically (in time)
foliated Schwarz symmetric, i.e., all elements in the associated omega-limit
set are axially symmetric with respect to a common axis passing through the
origin and are nonincreasing in the polar angle. We also obtain symmetry
results for solutions of H\'enon-type problems, for equilibria (i.e. for
solutions of the corresponding elliptic problem), and for time periodic
solutions.
