Abstract

We establish geometric regularity estimates for diffusive models driven by fully nonlinear second-order parabolic operators with measurable coefficients under a strong absorption condition as follows:

where $\Omega \subset {ℝ}^n$ is a bounded and smooth domain, $0\le \mu <1$ and ${\lambda }_0$ is bounded away from zero and infinity. Such models arise in applied sciences and become mathematically interesting because they permit the formation of dead-core zones, i.e., regions where nonnegative solutions vanish identically. Our main result gives sharp and improved $C^{2∕\left(1-\mu \right)}$ parabolic regularity estimates along the free boundary $\partial \left\{u>0\right\}$. In addition, we derive weak geometric and measure-theoretic properties of solutions and their free boundaries as: nondegeneracy, porosity, uniform positive density and finite speed of propagation. As an application, we prove a Liouville type result for entire solutions and we carry out a blow-up analysis. Finally, we prove the finiteness of parabolic $\left(n+1\right)$-Hausdorff measure of the free boundary for a particular class of operators.

Keywords

Mathematical Subject Classification 2010

Milestones

Authors