标题： 与分数$p$-拉普拉斯型算子相关的抛物方程解的局部有界性 显示英文标题

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

摘要： In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ 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 $(t;x,y)\in\R\times\R^d\times\R^d$. 与文献中现有的结果不同，本文获得的解的局部有界性扩展了线性情况下的已知结果（即$p=2$的情况），特别是使用了在所有$p\in (1,\infty)$上的时间$L^1$-范数的非局部抛物尾部。证明基于 De Giorgi-Nash-Moser 迭代中的新层次集截断以及迭代顺序的精心选择，以及一个通用的 Caccioppoli 类型不等式，该不等式被高效地应用于所有$p>1$的分数$p$-拉普拉斯型算子。 显示英文摘要

摘要： In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ 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 $(t;x,y)\in\R\times\R^d\times\R^d$. 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 $p>1$.