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

Abstract

In this paper, we study the boundedness and Holder continuity of local weak solutions to the following nonhomogeneous equation $$\begin{aligned} \partial _tu(x,t)+\mathrm{P.V.}\int _{{\mathbb {R}}^N}K(x,y,t)|u(x,t)-u(y,t)|^{p-2}\big (u(x,t)-u(y,t)\big )dy= f(x,t,u) \end{aligned}$$ in $$Q_T=\Omega \times (0,T)$$ , 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 Holder 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 Poincare Anal Non Lineaire, 2016).