Bifurcations on Hemispheres
Journal of Nonlinear Science.
1 (1991) 201-223.
As was pointed out by Crawford et al. [5], the source of this o nongenericity is that reaction diffusion equations are invariant under translations and reflections of the domain and, depending on boundary conditions, may naturally and uniquely be extended to larger domains with larger symmetry groups. These extra symmetries are the source of the nongenericity. In this paper we consider in detail the steady-state bifurcations of reaction diffusion equations defined on the hemisphere with Neumann boundary conditions along the equator. Such equations have a natural O(2) symmetry, but may be extended to the full sphere where the natural symmetry group is O(3). We also determine alarge class of partial differential equations and domains where this kind of extension is possible for both Neumann and Dirichlet boundary conditions.