On an evolutionary equation involving the nonlocal infinity Laplacian

We study the evolutionary equation $$\frac{\mathrm{\partial }u}{\mathrm{\partial }t}=({\mathcal{L}}_{\mathrm{\infty }}{)}^{\alpha }u\text{in}\mathrm{\Omega },$$ where $({\mathcal{L}}_{\mathrm{\infty }}{)}^{\alpha }u$ denotes a nonlocal infinity Laplacian acting on the function $u$, $0<\alpha \le 1$ and $\mathrm{\Omega }$ is a bounded open set in ${\mathbb{R}}^{n+1}$. We prove existence and uniqueness using Perron’s method for the Dirichlet problem when $\mathrm{\Omega }$ is a cylinder and $0<\alpha <1$.