Schultz [25] proved dispersive estimates for the wave equation on lattice graphs \(\mathbb {Z}^d\) for \(d=2,3,\) which was extended to \(d=4\) in [3]. By Newton polyhedra and the algorithm introduced by Karpushkin [16], 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 \(|t|^{-\frac{11}{6}}.\) Moreover, we prove Strichartz estimates and give applications to nonlinear equations.
