## Abstract

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ℝ^{N} × (-∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin ℝ^{N}. The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.

Original language | English (US) |
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Pages (from-to) | 1615-1638 |

Number of pages | 24 |

Journal | Communications in Partial Differential Equations |

Volume | 31 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2006 |

### Bibliographical note

Funding Information:This article is supported in part by NSF grant DMS-0400702.

## Keywords

- Quasilinear parabolic equations
- Symmetry of positive solutions
- Symmetry of unstable spaces

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