Symmetry of Positive Solutions of Asymptotically Symmetric Parabolic Problems on $${{\mathbb R}^N}$$

Abstract

In this paper we investigate symmetry properties of positive solution of quasilinear parabolic problems in the whole space. As the main result, we prove that if the problem converges exponentially to a symmetric one, then the solution converges to the space of symmetric functions. We also show, that this result does not hold true, if the convergence is not exponential.