(untitled) (Journal of Partial Differential Equations, 2006)
We consider a free boundary problem obtained from the asymptotic limit of a FitzHugh-Nagumo system, or more precisely, a slow-diffusion, fast-reaction equation governing a phase indicator, coupled with an ordinary differential equation governing a control variable ν. In the range (-1, 1), the v value controls the speed of the propagation of phase boundaries (interfaces) and in the mean time changes with dynamics depending on the phases. A new feature included in our formulation and thus made our model different from most of the contemporary ones is the nucleation phenomenon: a phase switch occurs whenever v elevates to 1 or drops to -1. For this free boundary problem, we provide a weak formulation which allows the propagation, annihilation, and nucleation of interfaces, and excludes interfaces from having (space- time) interior points. We study, in the one space dimension setting, the existence, uniqueness, and non-uniqueness of weak solutions. A few illustrating examples are also included.