Exponentially stable equilibria to an indefinite nonlinear Neumann problem in smooth domains

We prove existence of two nonconstant exponentially stable equilibria to the heat equation supplied with a nonlinear Neumann boundary condition in any smooth n-dimensional domain (n >= 2), independently of its geometry. The Neumann boundary condition reflects the fact that the flux on the boundary is proportional to the product of a prescribed bistable function of the density or concentration with an indefinite weight. Such solutions are obtained via variational methods, by minimizing the corresponding energy functional on suitable invariant sets to the semiflow generated by the parabolic problem. But this is possible only if the parameter in the boundary condition is sufficiently large, otherwise we prove using the Implicit Function Theorem the uniqueness of constant equilibrium solutions. The same theorem allows us to derive isolation and smooth dependence on the parameter for nonconstant exponentially stable equilibria found.