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

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.