Transversal and nontransversal intersections of stable and unstable manifolds in reaction diffusion equations on symmetric domains

Scalar reaction-diffusion equations on a ball in ℝ^N, N ≥ 2, with radially symmetric nonlinearities and Dirichlet boundary condition are considered. If the nonlinearity is nonincreasing in the radial variable (in particular if it is independent of it) it is proved that the stable and unstable manifolds of any two nonnegative equilibria intersect trasversally. The crucial property used in the proof is that the unstable manifold of a positive equilibrium consists of radially symmetric functions. Then, an equation is constructed that admits two radially symmetric equilibria whose invariant manifolds intersect nontransversally. In the appendix, examples of spatially homogeneous equations with positive equilibria with high Morse indices are given.