Nonuniqueness in a free boundary problem from combustion

We study a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. Vazquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion.