Hölder regularity for the parabolic perturbed fractional 1-Laplace equations

This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the decomposition of the nonlocal integral into local and nonlocal components to handle their contributions separately. We aim to show the local Hölder continuity of weak solutions within the parabolic domain. More precisely, the solutions are spatially $\alpha$-Hölder continuous with $0<\alpha <\min \{1,\frac{s_{p}p}{p-1}\}$ and $\gamma$-Hölder continuous in time, where the value of $\gamma$ is determined by the fractional differentiability indexes $s_1$, $s_p$ and the exponent $p$. For both the super-quadratic case ($p\ge 2$) and the sub-quadratic case ($1<p<2$), we establish the Sobolev regularity of solutions, which underpins the derivation of Hölder continuity. All estimates are quantitative and depend only on the structural parameters of the equation. To the best of our knowledge, this is the first attempt to develop a regularity theory for such nonlocal parabolic equations.