Fully-nonlinear parabolic dead-core problems

PABLO OCHOA; DA SILVA, JOÃO VITOR

PACIFIC JOURNAL OF MATHEMATICS

PACIFIC JOURNAL MATHEMATICS

Año: 2018

0030-8730

In this manuscript we establish geometric regularity estimates for diffusive models driven by fully non-linear second order parabolic operators with measurable coefficients under a strong absorption condition as followsegin{equation}label{eq1} mathcal{F}(x, t, D u, D^2 u) - partial_t u = lambda_{0}(x, t)u^{mu}chi_{{u>0}} quad mbox{in} quad Omega_Tdefeq Omega imes (0, T),end{equation}where $Omega subset R^n$ is a bounded and smooth domain, $0leq mu 0} $. In addiction, we derive weak geometric and measure theoretic properties of solutions and their free boundaries as: non-degeneracy, 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 $(n +1)$-Hausdorff measure of the free boundary for a particular class of operators.