Holder regularity for nonlocal in time subdiffusion equations with general kernel

TL;DR

This paper investigates the regularity of solutions to nonlocal in time subdiffusion equations with general kernels, establishing weak Harnack inequalities and Holder continuity, extending previous fractional derivative results.

Contribution

It introduces a broader class of kernels and inhomogeneities, extending regularity results for nonlocal subdiffusion equations beyond fractional derivatives.

Findings

Proved weak Harnack inequality for nonnegative weak supersolutions.

Established Holder continuity of weak solutions.

Extended regularity results to general PC kernels and inhomogeneous PDEs.

Abstract

We study the regularity of weak solutions to nonlocal in time subdiffusion equations for a wide class of weakly singular kernels appearing in the generalised fractional derivative operator. We prove a weak Harnack inequality for nonnegative weak supersolutions and Holder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work [12] by leaving the framework of distributed order fractional time derivatives and considering a general PC kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.