TY - JOUR

T1 - Symmetry properties of positive solutions of parabolic equations on ℝN

T2 - II. Entire solutions

AU - Poláčik, P.

PY - 2006/11/1

Y1 - 2006/11/1

N2 - We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ℝN × (-∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin ℝN. 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.

AB - We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ℝN × (-∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin ℝN. 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.

KW - Quasilinear parabolic equations

KW - Symmetry of positive solutions

KW - Symmetry of unstable spaces

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U2 - 10.1080/03605300600635020

DO - 10.1080/03605300600635020

M3 - Article

AN - SCOPUS:33750354303

VL - 31

SP - 1615

EP - 1638

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 11

ER -