Existence for nonlinear fractional p-Laplacian equations on finite graphs

다국어 초록 (Multilingual Abstract)

In this paper, we assume that $q>0$, $p>1$ and $s\in(0,1)$, and consider the following nonlinear fractional p-Laplacian equations on finite graphs: \begin{equation*} \left\{ \begin{array}{lll} \partial_t u^q(x,t)+(-\Delta)_p^su(x,t)=0,\\[2pt] u(x,t)|_{t=0}=u_0>0, \end{array} \right. \end{equation*} where $(-\Delta)_p^s$ is fractional Laplace operator on finite graphs. We establish the existence of solutions to the above parabolic equation using an iterative approach, which is different from previous works on graphs. Furthermore, we also derive some energy estimates of the solution. The difficulties lie in the estimates and the convergence of the nonlinear terms $\partial_t u^q(x,t)$ and $(-\Delta)^s_p u(x,t)$.