Threshold phenomena for a reaction-diffusion system

Abstract

AbstractWe consider the pure initial value problem for the system of equations νt = νxx + ƒ(ν) − w, wt= ε(ν − γw), ε, γ ⩾ 0, the initial data being (ν(x, 0), w(x, 0)) = (ϑ(x), 0). Here ƒ(v) = −v + H(v − a), where H is the Heaviside step function and a ϵ (0, 12). This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ϑ(x), is said to be superthreshold if limt→∞ s(t) = ∞. It is proven that the initial datum is superthreshold if ϑ(x) > a on a sufficiently long interval, ϑ(x) is sufficiently smooth, and ϑ(x) decays sufficiently fast to zero as ¦x¦ → ∞