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

P. Poláčik

doi:10.1080/03605300500299919

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

P. Poláčik

School of Mathematics

Research output: Contribution to journal › Article › peer-review

26 Scopus citations

Abstract

We consider quasilinear parabolic equations on ℝ^N 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 language	English (US)
Pages (from-to)	1567-1593
Number of pages	27
Journal	Communications in Partial Differential Equations
Volume	30
Issue number	11
DOIs	https://doi.org/10.1080/03605300500299919
State	Published - 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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10.1080/03605300500299919

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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.",

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author = "P. Pol{\'a}{\v c}ik",

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N2 - 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.

AB - 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.

KW - Asymptotic symmetry

KW - Cauchy problem

KW - Positive bounded solutions

KW - Quasilinear parabolic equations

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