Regularity of solutions for fully fractional parabolic equations

Abstract

In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -\Delta)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. First we derive H\"{o}lder 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 $$(\partial_t -\Delta)^{s} u(x,t) = b(x,t) |\nabla_x u (x,t)|^q + f(x, u(x,t))\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R}.$$ Under appropriate conditions, we prove that all nonnegative solutions, along with their spatial gradients, are uniformly bounded.