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.