Regularity for fully nonlinear degenerate parabolic equations with strong absorption

TL;DR

This paper proves sharp regularity and geometric properties for solutions to a class of fully nonlinear degenerate parabolic equations with strong absorption, extending previous results to the degenerate setting and introducing new estimates and principles.

Contribution

It establishes new regularity estimates, geometric properties, and comparison principles for degenerate parabolic equations with absorption, extending prior work to a broader degenerate context.

Findings

Sharp parabolic $C^{oldsymbol{eta}}$-regularity estimate at the free boundary.

Non-degeneracy and positive density of solutions.

Liouville-type theorem and gradient bounds for solutions.

Abstract

In this paper, we investigate dead-core problems for fully nonlinear degenerate parabolic equations with strong absorption, \begin{equation*} |Du|^{p} F(D^{2}u) - u_{t} = \lambda_{0}(x,t)\, u^{\mu}\, \chi_{\{u>0\}}(x,t) \qquad \text{in } \quad Q_{T} := Q \times (0,T), \end{equation*} where $0 \leq p < \infty$ and $0 < \mu < 1$. We establish a sharp and improved parabolic $C^{\alpha}$-regularity estimate along the free boundary $\partial \{ u > 0 \}$, where \[ \alpha := \frac{2+p}{1+p-\mu} > 1 + \frac{1}{1+p}. \] Moreover, we establish weak geometric properties of solutions, such as non-degeneracy and uniform positive density. As an application, we obtain a Liouville-type theorem for entire solutions and gradient bounds. Finally, as a byproduct of our approach, we derive a novel $L^{\delta}$-average estimate for fully nonlinear singular elliptic equations and present a new formulation of the gradient decay property. It is worth noting that the results presented here extend those in da Silva {\it et al.} ({\it Pacific J. Math}., \textbf{300} (2019), 179--213) and ({\it J. Differential Equations}., \textbf{264} (2018), 7270--7293) to the degenerate setting, and can be viewed as a parabolic analogue of da Silva {\it et al.} ({\it Math. Nachr}., \textbf{294} (2021), 38--55) and Teixeira ({\it Math. Ann}., \textbf{364} (2016), 1121--1134). Additionally, of independent mathematical interest, we emphasize that our manuscript establishes a comparison principle result and the compactness of viscosity solutions to fully nonlinear degenerate parabolic models with continuous and bounded forcing terms. These compactness and comparison properties serve as key ingredients in deriving enhanced regularity estimates along free boundary points for our model problem with strong absorption.