Higher Dimensional Bubble Profiles in a Sharp Interface Limit of the FitzHugh--Nagumo System

The FitzHugh-Nagumo system gives rise to a nonlocal geometric variational problem defined on subsets of a domain. The energy of a subset contains three terms: its perimeter, its volume, and a long-range self-interaction term represented by the integral of the solution to a screened Poisson's equation. A bubble profile is a ball-shaped stationary set when the domain is the entire space. If the space dimension is three or higher, depending on the parameters of the problem, there can be zero, one, or two bubble profiles. This is in contrast to an earlier result for the two-dimensional space, from which one may have three bubble profiles. The stability of each bubble is determined from the eigenvalues of the linearized operator. Using a stable bubble profile, one constructs a stationary assembly of perturbed balls on a general bounded domain, when the parameters are properly chosen.