Pointwise regularity for a parabolic equation with log-term singularity

Abstract

We obtained existence and pointwise regularity results for the following parabolic free boundary problem: $$\begin{aligned} u_t-\Delta u = \chi _{\{u>0\}} \log u \ \ \hbox {in} \ \ \Omega \times (0,T], \end{aligned}$$ with initial and boundary conditions in some appropriate spaces. The equation is singular along the set \(\partial \{u>0\}\), and the logarithmic nonlinearity does not have scaling properties. Thus, the machinery from regularity theory for free boundary problems, which strongly relies on the homogeneity of the problem, can not be applied directly. We prove that, near the free boundary, an approximate solution grows at most like \(r^2\log r.\) This is the so-called supercharacteristic growth, and its study has intriguing open questions. Our estimates are crucial to understand further analytic and geometric properties of the free boundary.