Journal of Dynamics and Differential Equations

, Volume 6, Issue 4, pp 639–658 | Cite as

Symmetrization properties of parabolic equations in symmetric domains

  • Anatoli V. Babin
Article

Abstract

Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD ⊂ R n 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.

Key words

Symmetry parabolic equations positive solutions stability 

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Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Anatoli V. Babin
    • 1
  1. 1.Moscow Institute of Engineers of Transportation, and School of MathematicsUniversity of MinnesotaMinneapolis

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