Autoregulated Impulse Point Heating of a Finite Medium

We consider a finite heat conducting medium whose boundary is maintained at zero temperature and, moreover, to which the same amount of heat is supplied at a certain point at the instant when the temperature at this point decreases to a given level. Up to an arbitrary shift in time, we prove the existence and uniqueness of a periodic regime with a unique heat pulse during each period. We present an efficient algorithm for constructing this regime if the medium is either an $n$-dimensional ball heated at the center or an interval heated at an arbitrary point.