Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

Abstract

In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $${\partial }_{t}u(t,x)-\text{p.v.}{\int }_{{\mathbb{R}}^d}|u(t,y)-u(t,x){|}^{p-2}(u(t,y)-u(t,x))J(t;x,y)dy=0,(t,x)\in \mathbb{R}\times {\mathbb{R}}^d,$$ where $\text{p.v.}$ means the integral in the principal value sense, $p\in (1,\infty )$ and $J(t;x,y)$ is comparable to the kernel of the fractional $p$-Laplacian operator $|x-y{|}^{-d-sp}$ with $s\in (0,1)$ and uniformly in $(t;x,y)\in \mathbb{R}\times {\mathbb{R}}^d\times {\mathbb{R}}^d$. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that $p=2$), in particular with a nonlocal parabolic tail that uses the $L^1$-norm in time for all $p\in (1,\infty )$. The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional $p$-Laplacian type operators with all $p>1$.

25 pages