Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime

2025/12/22 by Karthik Adimurthi, Mitesh Modasiya, Adimurthi, Karthik +1
Mathematics · #35A01 #35A15 #35K51 #35R11 #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.2512.19421

openalex publication_date 2025/12/22 · openalex created_date 2025/12/24 · openalex updated_date 2026/07/28

Abstract

We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form (|u|^q-2u)_t + P.V. ∫_ℝⁿ \frac|u(x,t) - u(y,t)|^p-2(u(x,t)-u(y,t))|x-y|^n+sp dy = 0, with p∈ (1,∞), q∈ (1,∞) and s ∈ (0,1). Analogous Hölder continuity result in the local case is known in the purely singular case \1

Citations

• Lipschitz regularity of fractional p-Laplacian
• Local Hölder regularity for bounded, signed solutions to nonlocal Trudinger equations
• Hölder regularity for parabolic fractional p-Laplacian
• Hölder regularity for fractional p-Laplace equations
• On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part III
• On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II
• On the Hölder regularity of signed solutions to a doubly nonlinear equation
• Continuity of solutions to a nonlinear fractional diffusion equation
• Fractional De Giorgi classes and applications to nonlocal regularity theory
• Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case
• Harnack estimates for quasi-linear degenerate parabolic differential equations
• Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation

Related

• Hölder regularity of weak solutions to nonlocal doubly degenerate parabolic equations
• Local Hölder regularity for nonlocal parabolic p-Laplace equations
• Hölder regularity for a class of doubly non linear PDEs
• Hölder regularity for mixed local and nonlocal p-Laplace parabolic equations
• Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem