Variational methods and implicit discrete Nagumo equation

Abstract

We analyze existence and uniqueness of ℓ2-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, at least for small time discretization steps h. Moreover, the solution is unique in the bistable case, λ>0.