A Parabolic Problem with a Fractional-Time Derivative

TL;DR

This paper investigates the regularity properties of solutions to a parabolic equation involving fractional derivatives in both space and time, establishing a Holder regularity theorem and results on existence, uniqueness, and higher regularity.

Contribution

It introduces a De Giorgi-Nash-Moser type regularity theorem for fractional parabolic equations in divergence form, along with existence, uniqueness, and higher regularity results.

Findings

Proved Holder regularity for solutions

Established existence and uniqueness of solutions

Demonstrated higher regularity in time

Abstract

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.