Vol. 300, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 304: 1
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Vol. 298: 1  2
Vol. 297: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
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:

(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.

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