The Wave Equation on Lattices and Oscillatory Integrals

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. Moreover, we notice that the Newton polyhedron can be used not only to interpret the decay rates for $d=2,3,4$, but also to study the most degenerate case for all odd $d\geq 3$. Furthermore, we prove $l^p\rightarrow l^q$ estimates as well as Strichartz estimates and give applications to nonlinear wave equations.