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We study the asymptotic (in time) behavior of positive and sign-changingsolutions to nonlinear parabolic problems in the whole space or in the exteriorof a ball with Dirichlet boundary conditions. We show that, under suitableregularity and stability assumptions, solutions are asymptotically (in time)foliated Schwarz symmetric, i.e., all elements in the associated omega-limitset are axially symmetric with respect to a common axis passing through theorigin and are nonincreasing in the polar angle. We also obtain symmetryresults for solutions of H\'enon-type problems, for equilibria (i.e. forsolutions of the corresponding elliptic problem), and for time periodicsolutions.

Symmetry properties of sign-changing solutions to nonlinear parabolic equations in unbounded domains