The Wave Equation on Lattices and Oscillatory Integrals

Cheng Bi, Jiawei Cheng, Bobo Hua

Published 2023-12-07Version 1

In this paper, we establish sharp dispersive estimates for the linear wave equation on the lattice $\mathbb{Z}^d$ with dimension $d=4$. Combining the singularity theory with results in uniform estimates of oscillatory integrals, we prove that the optimal time decay rate of the fundamental solution is of order $|t|^{-\frac{3}{2}}\log |t|$, which is the first extension of P. Schultz's results \cite{S98} in $d=2,3$ to the higher dimension. We also observe that the decay rate for $d=2,3,4$ can be well interpreted by the Newton polyhedra.