One-phase inverse Stefan problems with unknown nonlinear sources

We consider quasilinear models of inverse problems with phase transitions in a domain whose external boundary is a phase front with an unknown dependence on time. Additional information for finding the sources is given in the form of final overdetermination for the solution of the direct Stefan problem. On the basis of the duality principle, we obtain sufficient conditions for the uniqueness of the solution in Hölder classes for the considered inverse Stefan problems. We present examples in which the uniqueness property is lost when extending the set of admissible solutions.