Threshold Phenomena for a Reaction-Diffusion System

We consider the pure initial value problem for the system of equations vt = vxx + f(v)- w wt = (v- Yw),,y), 0, the initial data being (v(x,0),w(x,0)) = ('(x),O). Here f(v) =-v + H(v- a) 1 where H is the Heaviside step function and a e (o,-). 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, (x), is said to be superthreshold if lim s(t) = m It is proven that the initial datum is superthreshold if o(x)> a on a sufficiently long interval, p(x) is sufficiently smooth, and t(x) decays sufficiently fast to zero as 1xi. I " T., I-