TY - JOUR
T1 - Well-posedness of a Free Boundary Problem in the Limit of Slow-diffusion Fast-reaction Systems
JO - Journal of Partial Differential Equations
VL - 1
SP - 48
EP - 79
PY - 2006
DA - 2006/02
SN - 19
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5320.html
KW - Well-posedness
KW - FitzHugh-Nagumo system
KW - free boundary problem
AB - <p>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.</p>