On the existence of solutions to some singular parabolic free boundary problems

We construct nonnegative weak solutions to the singular parabolic free boundary problem $$\partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma ,$$ where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the right-hand side denotes the formal derivative of the non-smooth function $u \mapsto u_+^\gamma$. Weak solutions are obtained as limits of a suitable approximation procedure. We show uniform optimal regularity, optimal growth and nondegeneracy estimates, and a Weiss-type monotonicity formula for solutions to the approximating problem. Such uniform estimates are then passed to limit: we prove the existence of a class of weak solutions to the free boundary problem which is closed under blow-up and whose weak formulation encodes the sharp free boundary condition. Finally, we construct several examples of weak solutions with self-similar and traveling wave form.