Local boundedness and Hölder continuity for the parabolic fractional p-Laplace equations

In this paper, we study the boundedness and Hölder continuity of local weak solutions to the following nonhomogeneous equation ∂tu(x,t)+P.V.∫RNK(x,y,t)|u(x,t)-u(y,t)|p-2(u(x,t)-u(y,t))dy=f(x,t,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb {R}}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}\big (u(x,t)-u(y,t)\big )dy= f(x,t,u) \end{aligned}$$\end{document}in QT=Ω×(0,T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_T=\Omega \times (0,T)$$\end{document}, where the symmetric kernel K(x, y, t) has a generalized form of the fractional p-Laplace operator of s-order. We impose some structural conditions on the function f and use the De Giorgi-Nash-Moser iteration to establish the boundedness of local weak solutions in the a priori way. Based on the boundedness result, we also obtain Hölder continuity of bounded solutions in the superquadratic case. These results can be regarded as a counterpart to the elliptic case due to Di Castro et al. (Ann Inst H Poincaré Anal Non Linéaire, 2016).