H\"{o}lder Continuity for Fully Fractional Parabolic Equations with Space-time Nonlocal Operators

TL;DR

This paper proves local H"older continuity of solutions to fully fractional parabolic equations with space-time nonlocal operators, without assuming boundedness, by extending De Giorgi techniques to a nonlocal setting.

Contribution

It introduces a novel nonlocal De Giorgi iterative method to establish regularity results for fully fractional parabolic equations with space-time nonlocal operators.

Findings

Established local boundedness of solutions with tail terms.

Proved local H"older continuity of weak solutions.

Developed a nonlocal parabolic De Giorgi technique.

Abstract

We study the local H\"{o}lder regularity of weak solutions to the fully fractional parabolic equations involving spatial fractional diffusion and fractional time derivatives of the Marchaud type. It is worth noting that we do not impose boundedness assumptions on the weak solutions and nonhomogeneous terms. Within the space-time nonlocal framework, it is crucial to consider both space-dependent nonlocal tail terms and the first introduced time-dependent nonlocal tail term. By adapting a nonlocal variant of the parabolic De Giorgi iterative technique, we initially establish a priori local boundedness with tail terms for weak solutions and then prove the local H\"{o}lder continuity.