Symmetry of instabilities for scalar equations in symmetric domains (Q1903050)

Nonlinear second-order parabolic equations on a bounded domain under Dirichlet boundary conditions are considered. The domain and the equation are supposed to enjoy a reflectional symmetry. The following properties of positive solutions are discussed: Asymptotic symmetrization as \(t\to \infty\). Symmetry of the eigenfunctions corresponding to negative eigenvalues of the linearization at a positive equilibrium. Symmetry of the attractor for positive solutions. In the simplest example, \(u_t= \Delta u+ f(u)\) on a ball, one obtains the above properties with the full (radial) symmetry. Several references to articles where similar properties have been proved and/or applied are given. For more recent related results for nonautonomous equations see the papers of \textit{X.-Y. Chen} and the reviewer [J. Reine Angew. Math. 472, 17-51 (1996; Zbl 0839.35059)] and the author and \textit{G. R. Sell} [Attractors of nonautonomous parabolic equations and their symmetry properties (preprint)].