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

Schultz {S98} proved dispersive estimates for the wave equation on lattice graphs Zᵈ for $d=2,3,$ which was extended to $d=4$ in {BCH23}. By Newton polyhedra and the algorithm introduced by Karpushkin {K83}, we further extend the result to $d=5:$ the sharp decay rate of the fundamental solution of the wave equation on Z⁵ is |t|-11/6. Moreover, we prove Strichartz estimates and give applications to nonlinear equations.