Vol. 300, No. 1, 2019

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Fully nonlinear parabolic dead core problems

João Vítor da Silva and Pablo Ochoa

Vol. 300 (2019), No. 1, 179–213

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:

(x,t,Du,D2u) tu = λ0(x,t)uμχ {u>0} in ΩT := Ω × (0,T),

where Ω n is a bounded and smooth domain, 0 μ < 1 and λ0 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(1μ) parabolic regularity estimates along the free boundary {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 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.

dead-core problems, fully nonlinear parabolic equations, sharp and improved regularity estimates, parabolic Hausdorff measure estimates
Mathematical Subject Classification 2010
Primary: 35B65, 35K55
Received: 30 October 2017
Revised: 14 May 2018
Accepted: 9 September 2018
Published: 20 July 2019
João Vítor da Silva
Departamento de Matemática
Instituto de Ciências Exatas
Universidade de Brasília
Campus Universitário Darcy Ribeiro
Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS)
Ciudad Universitaria
Buenos Aires
Pablo Ochoa
Universidad Nacional de Cuyo-CONICET