Abstract
Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD ⊂ Rn are considered. It is assumed that the domainD and the equation are invariant with respect to a group {Q} of transformations ofD. In examples {Q} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact inC(D) negatively invariant set which consists of positive functions consists ofQ-symmetric functions. Examples of negatively invariant sets are (in autonomous case) equilibrium points, omega-limit sets, alpha-limit sets, unstable sets of invariant sets, and global attractors. Application of the main theorem to equilibrium points gives the Gidas-Ni-Nirenberg theorem. Applying the theorem to omega-limit sets, we obtain the asymptotical symmetrization property. That means that a bounded solutionu(t) asr→∞ approaches subspace of symmetric functions. One more result concerns properties of eigenfunctions of linearizations of the equations at positive equilibrium points. It is proved that all unstable eigenfunctions are symmetric.
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Babin, A.V. Symmetrization properties of parabolic equations in symmetric domains. J Dyn Diff Equat 6, 639–658 (1994). https://doi.org/10.1007/BF02218852
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DOI: https://doi.org/10.1007/BF02218852