Symmetry properties of positive solutions of parabolic equations on ℝ^N: II. Entire solutions

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.