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