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Symmetry Properties of Sign-Changing Solutions to Nonlinear Parabolic Equations in Unbounded DomainsWe study the asymptotic (in time) behavior of positive and sign-changing solutions to nonlin- ear 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 solu- tions of Hénon-type problems, for equilibria (i.e. for solutions of the corresponding elliptic problem), and for time periodic solutions.more » « less
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We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz ’63 model. Specifically, we prove that when damping is absent in one of the temperature components, the system possesses a unique invariant probability measure if and only if noise acts on the convection variable. On the other hand, if there is a positive growth term on the vertical temperature profile, we prove that there is no normalizable invariant state. Our approach relies on the derivation and analysis of nontrivial Lyapunov functions which ensure positive recurrence or null-recurrence/transience of the dynamics.more » « less
