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We study the symmetry properties of limit profiles of nonautonomous nonlinearparabolic systems with Dirichlet boundary conditions in radial bounded domains.In the case of competitive systems, we show that if the initial profilessatisfy a reflectional inequality with respect to a hyperplane, then all limitprofiles are foliated Schwarz symmetric with respect to antipodal points. Oneof the main ingredients in the proof is a new parabolic version of Serrin'sboundary point lemma. Results on radial symmetry of semi-trivial profiles arediscussed also for noncompetitive systems.

Qualitative properties of coexistence and semi-trivial limit profiles of nonautonomous nonlinear parabolic Dirichlet systems