Regularity for fully nonlinear degenerate parabolic equations with strong absorption

2025/12/09 by da Silva, João Vitor, Jiang, Feida, Wang, Jiangwen
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.2512.08196

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate dead-core problems for fully nonlinear degenerate parabolic equations with strong absorption, |Du|^p F(D²u) - u_t = λ₀(x,t) u^μ χ_\ugt;0\(x,t) in Q_T := Q × (0,T), where 0 ≤ p < ∞ and 0 < μ< 1. We establish a sharp and improved parabolic C^α-regularity estimate along the free boundary ∂ \ u > 0 \, where α:= (2+p)/(1+p-μ) gt; 1 + (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., 300 (2019), 179--213) and (\it J. Differential Equations., 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., 294 (2021), 38--55) and Teixeira (\it Math. Ann., 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.

Citations

• On the Grad-Mercier equation and Semilinear Free Boundary Problems
• Time derivative estimates for parabolic p-Laplace equations and applications to optimal regularity
• Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature
• Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and Hénon-type equations
• Interior W^2,δ type estimates for degenerate fully nonlinear elliptic equations with Lⁿ data
• Regularity of solutions for degenerate or singular fully nonlinear integro-differential equations
• Regularity of solutions to degenerate normalized p-Laplacian equation with general variable exponents
• Fully nonlinear dead-core systems
• Fully nonlinear dead-core systems
• On the fully nonlinear Alt-Phillips equation
• Reaction-diffusion equations for the infinity Laplacian
• Local regularity for quasi-linear parabolic equations in non-divergence form
• Regularity properties for p-dead core problems and their asymptotic limit as p → ∞
• Regularity properties for p−dead core problems and their asymptotic limit as p→∞
• Regularity for degenerate evolution equations with strong absorption
• Regularity for degenerate evolution equations with strong absorption
• Regularity for the fully nonlinear dead-core problem
• Sharp regularity estimates for second order fully nonlinear parabolic\n equations
• Infinity Laplacian equation with strong absorptions

Related

• Regularity for degenerate evolution equations with strong absorption
• C^{1, α}-regularity for solutions of degenerate/singular fully nonlinear parabolic equations
• Gradient regularity for widely degenerate parabolic equations
• Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and Hénon-type equations
• Sharp regularity estimates for second order fully nonlinear parabolic\n equations