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

TL;DR

This paper extends dispersive estimates for the wave equation to a 5-dimensional lattice graph, establishing sharp decay rates and applying these results to Strichartz estimates and nonlinear equations.

Contribution

It introduces a novel extension of dispersive estimates to five dimensions using Newton polyhedra, providing sharp decay rates and applications to nonlinear wave equations.

Findings

Sharp decay rate of |t|^{-11/6} for 5D lattice wave equation

Extension of dispersive estimates to 5D lattice graphs

Applications to nonlinear wave equations using Strichartz estimates

Abstract

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 $|t|^{-\frac{11}{6}}.$ Moreover, we prove Strichartz estimates and give applications to nonlinear equations.