Sharp time decay estimates for the discrete Klein-Gordon equation

TL;DR

This paper derives precise decay rates for solutions to the discrete Klein-Gordon equation on a cubic lattice in dimensions 2 to 4, improving previous conjectures and providing tools for nonlinear PDE analysis.

Contribution

It establishes sharp decay estimates for the discrete Klein-Gordon equation in multiple dimensions, surpassing earlier conjectures and introducing new analytical techniques.

Findings

Decay rate in 2D is |t|^{-3/4}

Decay rate in 3D is |t|^{-7/6}

Decay rate in 4D is |t|^{-3/2} log|t|

Abstract

We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d=2,3,4$. The $\ell^1\to\ell^{\infty}$ dispersive decay rate is $|t|^{-3/4}$ for $d=2$, $|t|^{-7/6}$ for $d=3$ and $|t|^{-3/2}\log|t|$ for $d=4$. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.