Symmetry properties of positive solutions of parabolic equations on ℝN: I. Asymptotic symmetry for the Cauchy problem

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Abstract

We consider quasilinear parabolic equations on ℝN satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic, center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.

Original languageEnglish (US)
Pages (from-to)1567-1593
Number of pages27
JournalCommunications in Partial Differential Equations
Volume30
Issue number11
DOIs
StatePublished - 2005

Bibliographical note

Funding Information:
The author is supported in part by NSF grant DMS-0400702.

Keywords

  • Asymptotic symmetry
  • Cauchy problem
  • Positive bounded solutions
  • Quasilinear parabolic equations

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