Abstract

Let be given positive constants and . For any (measurable) set with a volume and perimeter we study the geometric variational functional

 

 

where the nonlocal term is governed by in . This model comes as the -limit of the functional that gives rise to stationary FitzHugh-Nagumo system of equations. The critical points of is a set so that it is a sharp interface model with a nonlocal term. We give a survey of this model and its extension to cover traveling wave study.

 

For stationary cases we study when (i) and (ii) is the (flat) square torus (with periodic boundary conditions). For the same and in suitable regimes, in Case (i) multiple radially symmetric solutions with known exact multiplicity are found; while in Case (ii) planar and non-planar solutions co-exist by a bifurcation analysis, which also yields stability information of the non-planar solution.

 

Non-stationary FitzHugh-Nagumo equations admit traveling wave solutions with some specific wave speed . A similar -limit leads to a different functional whose critical point represents a set moving with a uniform seed in an infinite strip with periodic boundary condition in the -direction. Both the traveling wave shape and the speed need to be determined. Co-existence of planar traveling front, planar pulse and non-planar wave, all with distinct wave speeds, is established.

 

Organizers: Chueh-Hsin Chang (CCU), Chao-Nien Chen (NTHU), Chiun-Chuan Chen (NTU), Chih-Chiang Huang (CCU),Shyuh-yaur Tzeng (NCUE)