Fully nonlinear parabolic dead core problems

João Vítor da Silva and Pablo Ochoa

Vol. 300 (2019), No. 1, 179–213

DOI: 10.2140/pjm.2019.300.179

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:

$$\mathcal{ℱ}\left(x,t,Du,D^{2}u\right)-{\partial }_{t}u={\lambda }_0\left(x,t\right)u^{\mu }{\chi }_{\left\{u>0\right\}}\text{ in }{\Omega }_T:=\Omega \times \left(0,T\right),$$

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

dead-core problems, fully nonlinear parabolic equations, sharp and improved regularity estimates, parabolic Hausdorff measure estimates

Mathematical Subject Classification 2010

Primary: 35B65, 35K55

Milestones

Received: 30 October 2017

Revised: 14 May 2018

Accepted: 9 September 2018

Published: 20 July 2019