Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

Details Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

2011-10-07

arxiv.org/abs/1110.1543v2

Mathematics

Analysis of PDEs

16 pages

Scientific paper

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfy a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, i.e., all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e. solutions of the corresponding elliptic problem) and time periodic solutions.

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