Variational methods and implicit discrete Nagumo equation

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• 作者：Petr Stehlí ; k ; ^{pstehlik@kma.zcu.cz" class="auth_mail" title="E-mail the corresponding author} ; Joná ; &scaron ; Volek ^{volek1@kma.zcu.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Reaction&ndash ; diffusion equation ; Nagumo equation ; Variational methods ; Mountain pass geometry ; Implicit discretization
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：438
• 期：2
• 页码：643-656
• 全文大小：444 K

文摘

We analyze existence and uniqueness of ℓ²${\ell }^2$-solutions of the implicit discrete Nagumo reaction–diffusion equation. We study the infinite-dimensional problem variationally and describe corresponding potentials which have either the convex or mountain pass geometry. Consequently, we show that the implicit Nagumo equation has a solution for all reaction parameters λ∈R$\lambda \in \mathbb{R}$, at least for small time discretization steps h  . Moreover, the solution is unique in the bistable case, λ>0$\lambda >0$.