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Abstract
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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
is a bounded
and smooth domain,
and
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
parabolic regularity estimates along the free boundary
. 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
-Hausdorff
measure of the free boundary for a particular class of operators.
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Keywords
dead-core problems, fully nonlinear parabolic equations,
sharp and improved regularity estimates, parabolic
Hausdorff measure estimates
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Mathematical Subject Classification 2010
Primary: 35B65, 35K55
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Milestones
Received: 30 October 2017
Revised: 14 May 2018
Accepted: 9 September 2018
Published: 20 July 2019
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