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 u_t = D u_{x x} + e^{- 1 / u} - σ (u - α), with u_x (0, t) = u_x (1, t) = 0, on I = [0, 1]. They are composed from the positive non-constant solutions of D v_{x x} + e^{- 1 / v} - σ (v - α) = 0, with v_x (0) = v_x (1) = 0, for small values of D. We will give easily verifiable conditions for when combustion waves arise and when they do not.