Volume 19, Issue 1
Well-posedness of a Free Boundary Problem in the Limit of Slow-diffusion Fast-reaction Systems

Xinfu Chen & Congyu Gao

DOI:

J. Part. Diff. Eq., 19 (2006), pp. 48-79.

Published online: 2006-02

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  • Abstract

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.

  • Keywords

Well-posedness FitzHugh-Nagumo system free boundary problem

  • AMS Subject Headings

35R35.

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COPYRIGHT: © Global Science Press

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@Article{JPDE-19-48, author = {}, title = {Well-posedness of a Free Boundary Problem in the Limit of Slow-diffusion Fast-reaction Systems}, journal = {Journal of Partial Differential Equations}, year = {2006}, volume = {19}, number = {1}, pages = {48--79}, abstract = {

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.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5320.html} }
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 -

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.

Xinfu Chen & Congyu Gao . (2019). Well-posedness of a Free Boundary Problem in the Limit of Slow-diffusion Fast-reaction Systems. Journal of Partial Differential Equations. 19 (1). 48-79. doi:
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