Sharp dispersive estimates for the wave equation on the 5-dimensional lattice graph

Schultz \cite{S98} proved dispersive estimates for the wave equation on lattice graphs ${\mathbb{Z}}^d$ for $d=2,3,$ which was extended to $d=4$ in \cite{BCH23}. By Newton polyhedra and the algorithm introduced by Karpushkin \cite{K83}, we further extend the result to $d=5:$ the sharp decay rate of the fundamental solution of the wave equation on ${\mathbb{Z}}^5$ is $\mathrm{\mid }t{\mathrm{\mid }}^{-\frac{11}{6}}\mathrm{.}$ Moreover, we prove Strichartz estimates and give applications to nonlinear equations.