Collapsing heat waves

In certain combustion models, an initial temperature profile will develop into a combustion wave that will travel at a specific wave speed. Other initial profiles do not develop into such waves, but die out to the ambient temperature. There exists a clear demarcation between those initial conditions that evolve into combustion waves and those that do not. This is sometimes called a watershed initial condition. In this paper we will show that there may be numerous exact watershed conditions to the initial–Neumann–boundary value problem ut=Duxx+e−1/u−σ(u−α), with ux(0,t)=ux(1,t)=0ux(0,t)=ux(1,t)=0, on I=[0,1]I=[0,1]. They are composed from the positive non-constant solutions of Dvxx+e−1/v−σ(v−α)=0, with vx(0)=vx(1)=0vx(0)=vx(1)=0, for small values of DD. We will give easily verifiable conditions for when combustion waves arise and when they do not.