Elliptic regularization of some semilinear parabolic free boundary problems

Abstract

We prove existence of strong solutions to a family of some semilinear parabolic free boundary problems by means of elliptic regularization. Existence of solutions is obtained in two steps: we first show some uniform energy estimates and then we pass to the weak limit. To carry out this second step, we establish uniform non-degeneracy estimates for the approximating sequence as well as parabolic non-degeneracy and optimal regularity for the limit solution. To the best of our knowledge, this is the first time the elliptic regularization approach is used in the context of parabolic obstacle problems.