#### Vol. 300, No. 1, 2019

 Recent Issues Vol. 307: 1  2 Vol. 306: 1  2 Vol. 305: 1  2 Vol. 304: 1  2 Vol. 303: 1  2 Vol. 302: 1  2 Vol. 301: 1  2 Vol. 300: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Contacts Submission Guidelines Submission Form Policies for Authors ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Special Issues Author Index To Appear Other MSP Journals
Fully nonlinear parabolic dead core problems

### João Vítor da Silva and Pablo Ochoa

Vol. 300 (2019), No. 1, 179–213
##### 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
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
##### Authors
 João Vítor da Silva Departamento de Matemática Instituto de Ciências Exatas Universidade de Brasília Campus Universitário Darcy Ribeiro Brasília Brazil Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS) (CONICET-Argentina) Ciudad Universitaria Buenos Aires Argentina Pablo Ochoa Universidad Nacional de Cuyo-CONICET Mendoza Argentina