FULLY NONLINEAR PARABOLIC DEAD CORE PROBLEMS

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: F(x, t, D(2)u) - partial derivative(t)u = lambda(0(x, t)u mu) chi({u>0}) in Omega(T) := Omega x (0, T), where Omega subset of R-n is a bounded and smooth domain, 0 <= 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 C2/(1-mu) parabolic regularity estimates along the free boundary partial derivative{u > 0}. 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 Lionville 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.