Regularity of solutions for fully fractional parabolic equations

TL;DR

This work develops local regularity theory for the fully fractional parabolic equation $(\partial_t-\Delta)^s u=f$ with $0<s<1$ and nonnegative forcing. By introducing a directional perturbation average for the fully fractional heat kernel and representing solutions via Green’s function, the authors derive Hölder and Schauder estimates for both the homogeneous and nonhomogeneous parts, even when global bounds are unavailable. They successfully replace global $L^{\infty}$ data with local bounds to control higher norms, enabling blow-up and rescaling analyses on unbounded domains and leading to a priori estimates for nonlinear master equations, including gradient-driven nonlinearities. The results generalize regularity theory for nonlocal parabolic problems and provide tools applicable to a broad class of unbounded-domain equations, with implications for existence, uniqueness, and qualitative behavior of solutions.

Abstract

In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -Δ)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. First we derive Hölder and Schauder estimates for nonnegative solutions of this equation. Due to the {\em nonlocality} of the master operator, existing results (cf. \cite{ST}) rely on global bounds of the solutions $u$ to control their higher local norms. However, such results are inadequate for blow-up and rescaling analysis aimed at obtaining a priori estimates for solutions to {\em nonlocal } equations on unbounded domains, as the global norms of the rescaled functions may diverge. This limitation raises to a natural and challenging question: {\em Can local bounds of solutions replace global bounds to control their higher local norms?} Here, we provide an affirmative answer to this question for nonnegative solutions. To achieve this, we introduced several new ideas and novel techniques. One of the key innovations is to use a {\em directional perturbation average} to derive an important estimate for the fully fractional heat kernel, as stated in Lemma \ref{key0}. We believe this estimate, along with other new techniques introduced here, will serve as powerful tools in regularity estimates for a wide range of nonlocal equations. Building on this breakthrough, we employ the blow-up and rescaling arguments to establish a priori estimates for solutions to a broader class of nonlocal equations in unbounded domains, such as $$(\partial_t -Δ)^{s} u(x,t) = b(x,t) |\nabla_x u (x,t)|^q + f(x, u(x,t))\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R}.$$ Under appropriate conditions, we prove that all nonnegative solutions, along with their spatial gradients, are uniformly bounded.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Theorem 4
• Remark 2
• Lemma 1: Fully fractional heat kernel estimates
• Theorem 5
• Theorem 6
• Remark 3