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

TL;DR

This paper establishes the existence of weak solutions to a class of singular parabolic free boundary problems with non-smooth nonlinearities, providing regularity, growth, and nondegeneracy estimates, and constructing explicit solution examples.

Contribution

It introduces an approximation method to prove existence of solutions, derives uniform estimates, and constructs explicit self-similar and traveling wave solutions.

Findings

Existence of weak solutions via approximation methods.

Uniform regularity and growth estimates for solutions.

Construction of explicit self-similar and traveling wave solutions.

Abstract

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.