1. The temperature is quickly raised at the left end of the sample from zero up to some value while the right end is thermally isolated. The heat propagates from left to right, then the thermal runaway (ignition) occurs somewhere, then the deflagration front propagates from the ignition point, resulting in a complete burning

a) W_f=3. The temperature is raised up to the flame temperature T_f on the left. In this case the runaway occurs immediately at the left end and the front moves to the right:

Temperature, Metastable population

b) W_f=3. The temperature is raised up to 0.3T_f on the left. The runaway occurs after ignition time at some distance from the left end:

Temperature, Metastable population

c) W_f=10 - deflagration is slower, ignition time is longer. Thus the heat penetrates deeper into the sample before ignition occurs. The temperature is raised up to 0.55T_f on the left, ignition occurs around x=15:

Temperature, Metastable population

d) W_f=10. The temperature is raised up to 0.5T_f on the left, ignition occurs at the right end:

Temperature, Metastable population

2. In the initial state the sample is locally overheated in the middle with a Gaussian temperature profile of some height and some width. The temperature at the boundaries is zero. Then the system evolves in time. At first the heat spreads to left and right, the height of the temperature profile decreases and the width increases. On the other hand, the heat is released in the middle of the sample because of burning that competes with heat diffusion. There are two scenarios: (i) Subcritical: The heat diffusion prevails and the local overheating disappears with time; (ii) Supercritical: The heat release prevails, ignition occurs in the middle, and the deflagration front propagates to the ends. What happens depends on the height and width of the initial temperature profile, as well as on other parameters. Below the slightly supercritical scenario is shown:

Temperature, Metastable population