Hölder regularity for nonlocal in time subdiffusion equations with general kernel

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 Hölder continuity of weak solutions to such problems. Our results substantially extend the results from our previous work [11] by leaving the framework of distributed order fractional time derivatives and considering a general $\mathrm{PC}$ kernel and by also allowing for an inhomogeneity in the PDE from a Lebesgue space of mixed type.