Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

We consider the Dirichlet problem for a class of semilinear parabolic equations on a bounded domain which is reflectionally symmetric about a hyperplane H. The equations consist of a symmetric time-autonomous part and a nonsymmetric perturbation which decays to zero as time approaches infinity. In our first theorem, we prove the asymptotic symmetry of each bounded positive solution of this asymptotically symmetric problem. The novelty of this result is that the solutions considered are not assumed uniformly positive, which prevents one from applying common techniques based on the method of moving hyperplanes. In our second main theorem, we classify the positive entire solutions of the unperturbed time-autonomous problems. In particular, we characterize all entire solutions, which are not symmetrically decreasing in the direction orthogonal to H, as connecting orbits from an equilibrium with a nontrivial nodal set to another invariant set.