# Symmetrization properties of parabolic equations in symmetric domains

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## Abstract

Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domain*D* ⊂ R^{ n } are considered. It is assumed that the domain*D* and the equation are invariant with respect to a group {*Q*} of transformations of*D*. In examples {*Q*} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact in*C(D)* negatively invariant set which consists of positive functions consists of*Q*-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 solution*u(t)* as*r*→∞ 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.

### Key words

Symmetry parabolic equations positive solutions stability## Preview

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