Regularity for fully nonlinear degenerate parabolic equations with strong absorption
João Vitor da Silva, Feida Jiang, Jiangwen Wang
TL;DR
The paper develops a comprehensive geometric regularity theory for dead-core solutions to fully nonlinear degenerate parabolic equations with strong absorption in the form $|Du|^{p}F(D^{2}u)-u_t=\lambda_{0}(x,t)u^{\mu}\chi_{\{u>0\}}$, where $0<\mu<1$ and $p\ge0$. Through intrinsic scaling, barrier constructions, and viscosity-solution techniques, it establishes sharp $C^{\alpha}$ regularity along the free boundary with $\alpha=\frac{2+p}{1+p-\mu}$, along with non-degeneracy, positive density, finite propagation speed, and porosity of the free boundary; it also proves Liouville-type theorems and gradient-decay estimates, and introduces a new $L^{\delta}$-average framework for related elliptic problems. The work extends previous results to the degenerate parabolic setting and provides a robust compactness and comparison theory for fully nonlinear degenerate parabolic models, enriching the mathematical understanding of dead-core phenomena in diffusion-reaction processes. Collectively, these results offer a rigorous dead-core theory with potential applications to reaction-diffusion and phase-transition models, and they pave the way for further extensions to more general degenerate operators and systems.
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} = λ_{0}(x,t)\, u^μ\, χ_{\{u>0\}}(x,t) \qquad \text{in } \quad Q_{T} := Q \times (0,T), \end{equation*} where $0 \leq p < \infty$ and $0 < μ< 1$. We establish a sharp and improved parabolic $C^α$-regularity estimate along the free boundary $\partial \{ u > 0 \}$, where \[ α:= \frac{2+p}{1+p-μ} > 1 + \frac{1}{1+p}. \] 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 $L^δ$-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.